Then the molar mass of air is computed by M0 = R/Rair = 28.964917g/mol. Or. In statistical mechanics, it can be proven 2. R = 8.314 kPa L / (K mol) = How do you calculate the molar mass of a gas? statistical-mechanics. Chemistry: Why This Is Important: Ideal Gases, The Kinetic Molecular Theory of Gases: Why Gases Do What They Do, Chemistry: The Kinetic Molecular Theory of Gases, The Washington Monument: Facts, History & Profile. Gay-Lussac's law has a constant volume. When purchasing wardrobe lighting systems it's essential to carefully consider which types and features will best complement the size, shape, and electrical connection of your space; your closet size, shape, and electricity availability all play into what type of . Direct link to lisa_cassaniti's post I know that Charles Law n, Posted 2 years ago. R is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant, In this equation the symbol R is a constant called the universal gas constant that has the same value for all gasesnamely, R = 8.31 J/mol K. The power of the ideal gas law is in its simplicity. Compressibility Factor. My guess is that the entropy of the nonideal gas should be greater. Upper Saddle River: Pearson Education, Inc., 2007. Here the G has both the purpose by taking the value Through advanced mathematics (provided in outside link if you are interested), the properties of the three simple gas laws will give you the Ideal Gas Equation. 6.674 08 x 10-11m3kg-1s-2 it is giving up the exact force which when two masses of 1 kg each will exert on each other when kept 1 m apart. The formula of the gas constant from the ideal gas law equation is. Imagine that you have a thermos bottle filled with a gas having a piston at its top which you can pull/push, an electric resistance inside that you can use to heat the gas, a thermometer and a barometer. Learn more about Stack Overflow the company, and our products. The greater it deviates from the number 1, the more it will behave like a real gas rather than an ideal. Some say the symbol for the gas constant is named in honour of French chemist Henri Regnault. Attempt them initially, and if help is needed, the solutions are right below them. Accessibility StatementFor more information contact us atinfo@libretexts.org. on weid properties of melting ice. Why is water a good solvent for recrystallization. . . 3 R is simply the ratio of the pressure and volume to the moles of gas and temperature. Physical constant equivalent to the Boltzmann constant, but in different units, Measurement and replacement with defined value, "Ask the Historian: The Universal Gas Constant Why is it represented by the letter, D. Mendeleev. It is a proportionality constant for the ration of #(PV)/(nT)#,where P is pressure, V is volume, n is moles of the gas, and T is the temperature in Kelvin. As students, professors, and chemists, we sometimes need to understand the concepts before we can apply it, and assuming the gases are in an ideal state where it is unaffected by real world conditions will help us better understand the behavior the gases. One of the most important formulas in thermodynamics is P1 * V1 / T1= P2 * V2 / T2. The origin of the symbol R for the ideal gas constant is still obscure. However, they had encountered many difficulties because of the fact that there always are other affecting factors such as intermolecular forces. ", Luder, W. F. "Ideal Gas Definition." Pressure is directly proportional to number of molecule and temperature. (Since P is on the same side of the equation with V), The universal value of STP is 1 atm (pressure) and 0. Step 1: Write down your given information, \[(248 \; \rm{Torr}) \times \dfrac{1 \; \rm{atm}}{760 \; \rm{Torr}} = 0.3263 \; \rm{atm}\]. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Constants in physics are not just unit matching things. @ShawnO'Brien Boltzmann's constant (or the gas constant) is just an arbitrary conversion between energy and temperature. \[\dfrac{P}{n_{Ne}} = \dfrac{P}{n_{CO_2}}\], \[\dfrac{1.01 \; \rm{atm}}{0.123\; \rm{mol} \;Ne} = \dfrac{P_{CO_2}}{0.0144\; \rm{mol} \;CO_2} \], \[P_{total}= 1.01 \; \rm{atm} + 0.118\; \rm{atm}\], \[P_{total}= 1.128\; \rm{atm} \approx 1.13\; \rm{atm} \; \text{(with appropriate significant figures)} \]. Ultimately, the reason is that the atoms of an ideal gas are non-interacting point particles. What is an "ideal gas"? Direct link to Michelle Chen's post When converting, why shou, Posted 4 years ago. Volume is not a variable in his formula. ( P + a n 2 V 2) ( V n b) = n R T. It fits pressure-volume-temperature data for a real gas better than the ideal gas equation does. ], [Could we have used the other gas constant? Direct link to Matt B's post No calculus needed :-) Li, Posted 7 years ago. K 1) T = temperature in Kelvin. This answer contained what I believed to be several errors all related to confusion about the difference between units and dimensions. 5.0 g of neon is at 256 mm Hg and at a temperature of 35 C. What is the volume? the pressure-volume product, rather than energy per temperature increment per particle. Ideal Gas Constant: Definition, Values, and Units - ChemistryGod Volume of a gas is directly proportional to the amount of gas at a constant temperature and pressure. Other things to keep in mind: Know what Standard Temperature and Pressure (STP) values are. Since you can't divide by 0, the formula would not work. { "Avogadro\'s_Law" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Boyle\'s_Law" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Charles\'s_Law_(Law_of_Volumes)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Dalton\'s_Law_(Law_of_Partial_Pressures)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Gas_Laws:_Overview" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", The_Ideal_Gas_Law : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { Chemical_Reactions_in_Gas_Phase : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Gases_(Waterloo)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Gas_Laws : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Gas_Pressure : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Kinetic_Theory_of_Gases : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Properties_of_Gas : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Real_Gases : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "showtoc:no", "license:ccbyncsa", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FSupplemental_Modules_(Physical_and_Theoretical_Chemistry)%2FPhysical_Properties_of_Matter%2FStates_of_Matter%2FProperties_of_Gases%2FGas_Laws%2FThe_Ideal_Gas_Law, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Standard condition of temperature and pressure is known as, Take note of certain things such as temperature is always in its, the particles have no forces acting among them, and. Now we can plug these variables into our solved version of the molar ideal gas law to get, Now to determine the number of air molecules. The gas constant (cried the molar, universal, or ideal gas constant an aa, denotit bi the seembol R or R) is a pheesical constant which is featurt in mony fundamental equations in the pheesical sciences, such as the ideal gas law an the Nernst equation. thermodynamics - Why is there a constant in the ideal gas law Direct link to Matt B's post You are right, the R actu, Posted 5 years ago. The ideal gas law is an "equation of state" that describes the relationship between pressure (#P#), density (#n/V#) and temperature (#T#). Is the Boltzmann constant really that important? They are actually very fundamental. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. If, however, you like pressures in atmospheres and volumes in Liters, well then your gas constant is going to be 0.0821 Latm/molK. A \(0.633\;\rm{g}\) sample of \(CO_2\) vapor is then added. It's very difficult to come up with rules for describing the behaviors of real gases because they come in a variety of different shapes and sizes, as well as experience different intermolecular forces to various degrees. A. collide more frequently with each other. "China Is Killing Americans!" - Reaction To Xi Jinping Unifying U.S.'s Biggest Enemies. Some common values for [math]R[/math] are: This constant is closely related to Boltzmann's constant, [math]k_B[/math] (to get the ideal gas constant from Boltzmann's constant, multiply by Avogadro's number: [math]R=N_A k_B [/math]). An equation due to van der Waals extends the ideal gas equation in a straightforward way. Definition: Gas constant is the general constant in an equation of a gaseous state which is equivalent to the product of the pressure and volume of one mole divided by absolute temperature. 1 minute = 60 seconds. mol) T equals the temperature measured in Kelvin. For this reason, many students are taught the three most important gas laws by . Alternatively, we could have solved this problems by using the molecular version of the ideal gas law with Boltzmann's constant to find the number of molecules first, and then converted to find the number of moles. The ideal gas law is -. All rights reserved including the right of reproduction in whole or in part in any form. Lastly, the constant in the equation shown below is R, known as the the gas constant, which will be discussed in depth further later: Another way to describe an ideal gas is to describe it in mathematically. An ideal gas will always equal 1 when plugged into this equation. It is used to determine the rate constant k. where A is the Arrhenius constant and Ea is the activation energy. $$pV=T \tag{2}$$. As you can see in (4) the units of $pV$ turns out to be $J$. No calculus needed :-) Like most any constants, they are simply needed if there is always that same factor missing in an equation. On the other hand, it is an arbitrary application of a mathematical expression to fit experimental observations quite possibly devoid of any basis in physical reality and can therefore easily fail to predict behavior outside of very narrow ranges of applicability. I was using the term 'units' to refer to both scale and dimensionality, which is a common way to speak. What volume (L) will 0.20 mol HI occupy at 300 K and 100.0 kPa? A few things should always be kept in mind when working with this equation, as you may find it extremely helpful when checking your answer after working out a gas problem. The gas constant (R) is a proportionality constant used in the ideal gas law and Nernst equation. 2.12: Van der Waals' Equation - Chemistry LibreTexts What is the density of nitrogen gas (\(N_2\)) at 248.0 Torr and 18 C? Ideal gas | Definition, Equation, Properties, & Facts | Britannica It is only important if you want to relate the pressure or the volume or the moles or the temperature of a gas to any of the other values. If there is Ideal Gas constant, then do we have real gas constant? Direct link to The #1 Pokemon Proponent's post That is the definition of, Posted 7 years ago. introduction. molecules) of a gas remains the same, the quantity, This formula is particularly useful when describing an ideal gas that changes from one state to another. mol-1). or expressed from two pressure/volume points: This equation would be ideal when working with problem asking for the initial or final value of pressure or volume of a certain gas when one of the two factor is missing. Dead Island 2 Full Game on Ps5 | zombie apocalyptic fiction Can someone explain why this point is giving me 8.3V? Now we can generate an universal value for $_0$ as, $$_0=R=\frac{p_0 V_0}{T_0}=\frac{101.325 10^522.410^{-3} \, \mathrm{\frac{N}{m^2}m^3}}{273.15 \, \mathrm{K}}=8.3 \, \mathrm{J/K} \tag{4}$$. where: P is the pressure exerted by an ideal gas, V is the volume occupied by an ideal gas, T is the absolute temperature of an ideal gas, R is universal gas constant or ideal gas constant, n is the number of moles (amount) of gas.. Derivation of Ideal Gas Law. R = is the universal gas constant = 8.3145 J/mol K. N = is the number of molecules. Counting and finding real solutions of an equation. But there is more in (5) then just a compact form of describing the thermodynamics system. If you are using liters and atmospheres of pressure, instead of Pascals and cubic meters, then you have the following: P equals pressure measured in atmospheres. The gas constant has the same unit as of entropy and molar heat capacity. In this issue, two well-known assumptions should have been made beforehand: An ideal gas is a hypothetical gas dreamed by chemists and students because it would be much easier if things like intermolecular forces do not exist to complicate the simple Ideal Gas Law. If you happen to use newtons as your pressure and m3 as . 1.5.4.2 Ideal Gas Theory. If you use the gas constant. Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Enthalpy Change in Reversible, Isothermal Expansion of Ideal Gas. Consider the following equation: The term \(\frac{pV}{nRT}\) is also called the compression factor and is a measure of the ideality of the gas. The theory behind the ideal gas law is that gas molecules undergo perfectly elastic (kinetic energy-conserving) collisions in a container of fixed volume, in which they take up none of the available space. Now for gas constant ($R$): it is an experimental constant. Remark: The units must cancel out to get the appropriate unit; knowing this will help you double check your answer. Ideal gases are essentially point masses moving in constant, random, straight-line motion. B) It is a combustible gas. D) It has a boiling point of -252.87C. What volume (L) will 0.20 mol HI occupy at 300 K and 100.0 kPa? @DanielSank But it still a mistake confusing temperature and energy. It only takes a minute to sign up. They are like symmetry points were everything moving around most do so in a way to keep their values the same. When should I use the ideal gas law and not the combined gas law? The ideal gas constant and the Boltzmann constant (kB) are related by Avogadro's constant (NA). \[n_{CO_2} = 0.633\; \rm{g} \;CO_2 \times \dfrac{1 \; \rm{mol}}{44\; \rm{g}} = 0.0144\; \rm{mol} \; CO_2\]. and the first example, shouldn't the atm version of the ideal gas constant be 0.082 L*atm/mol*K instead of 0.082 L*atm/K? What is the physical significance of the universal gas constant R?Why is the internal energy of a real gas a function of pressure and \[ V = \dfrac{(0.25\; \rm{mol})(0.08206\; \rm{L atm}/\rm{K mol})(308\; \rm{K})}{(0.3368\; \rm{atm})}] \]. Note that for the case of the ideal gas law, it would be perfectly okay to write $PV = NT$; you would just have to understand that $T$ now means something different, i.e. What's the cheapest way to buy out a sibling's share of our parents house if I have no cash and want to pay less than the appraised value? The Ideal Gas Law is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Direct link to RandomDad's post Where do _R, Na(Avogadro', Posted 7 years ago. "Gas constant," Wikipedia, 2021. minus, 200, start text, space, C, end text, R, equals, 8, point, 31, start fraction, J, divided by, K, dot, m, o, l, end fraction, start text, p, a, s, c, a, l, s, space, end text, P, a, start text, k, e, l, v, i, n, space, end text, K, R, equals, 0, point, 082, start fraction, L, dot, a, t, m, divided by, K, dot, m, o, l, end fraction, start text, a, t, m, o, s, p, h, e, r, e, s, space, end text, a, t, m, start text, l, i, t, e, r, s, space, end text, L, N, start text, space, n, u, m, b, e, r, space, o, f, space, m, o, l, e, c, u, l, e, s, end text, n, start text, space, m, o, l, e, s, end text, P, V, equals, N, k, start subscript, B, end subscript, T, k, start subscript, B, end subscript, equals, 1, point, 38, times, 10, start superscript, minus, 23, end superscript, start fraction, J, divided by, K, end fraction, start text, p, a, s, c, a, l, s, space, P, a, end text, start text, k, e, l, v, i, n, space, K, end text, n, R, equals, N, k, start subscript, B, end subscript, equals, start fraction, P, V, divided by, T, end fraction, equals, start text, space, c, o, n, s, t, a, n, t, end text, start fraction, P, V, divided by, T, end fraction, start fraction, P, start subscript, 1, end subscript, V, start subscript, 1, end subscript, divided by, T, start subscript, 1, end subscript, end fraction, equals, start fraction, P, start subscript, 2, end subscript, V, start subscript, 2, end subscript, divided by, T, start subscript, 2, end subscript, end fraction, 1, point, 54, start text, space, a, t, m, end text, 0, point, 119, start text, space, m, end text, 25, start superscript, o, end superscript, start text, space, C, end text, (usethemolarformoftheidealgaslaw), P, V, equals, n, R, T, start text, left parenthesis, u, s, e, space, t, h, e, space, m, o, l, a, r, space, f, o, r, m, space, o, f, space, t, h, e, space, i, d, e, a, l, space, g, a, s, space, l, a, w, right parenthesis, end text, n, equals, start fraction, P, V, divided by, R, T, end fraction, start text, left parenthesis, s, o, l, v, e, space, f, o, r, space, t, h, e, space, n, u, m, b, e, r, space, o, f, space, m, o, l, e, s, right parenthesis, end text, (decidewhichgasconstantwewanttouse), n, equals, start fraction, P, V, divided by, left parenthesis, 8, point, 31, start fraction, J, divided by, K, dot, m, o, l, end fraction, right parenthesis, T, end fraction, start text, left parenthesis, d, e, c, i, d, e, space, w, h, i, c, h, space, g, a, s, space, c, o, n, s, t, a, n, t, space, w, e, space, w, a, n, t, space, t, o, space, u, s, e, right parenthesis, end text, start text, p, a, s, c, a, l, s, end text, 1, point, 54, start text, space, a, t, m, end text, times, left parenthesis, start fraction, 1, point, 013, times, 10, start superscript, 5, end superscript, start text, space, P, a, end text, divided by, 1, start text, space, a, t, m, end text, end fraction, right parenthesis, equals, 156, comma, 000, start text, space, P, a, end text, start fraction, 4, divided by, 3, end fraction, pi, r, cubed, V, equals, start fraction, 4, divided by, 3, end fraction, pi, r, cubed, equals, start fraction, 4, divided by, 3, end fraction, pi, left parenthesis, 0, point, 119, start text, space, m, end text, right parenthesis, cubed, equals, 0, point, 00706, start text, space, m, end text, cubed, T, start subscript, K, end subscript, equals, T, start subscript, C, end subscript, plus, 273, start text, space, K, end text, T, equals, 25, start superscript, o, end superscript, start text, space, C, end text, plus, 273, start text, space, K, end text, equals, 298, start text, space, K, end text, (plugincorrectunitsforthisgasconstant), n, equals, start fraction, left parenthesis, 156, comma, 000, start text, space, P, a, end text, right parenthesis, left parenthesis, 0, point, 00706, start text, space, m, end text, cubed, right parenthesis, divided by, left parenthesis, 8, point, 31, start fraction, J, divided by, K, dot, m, o, l, end fraction, right parenthesis, left parenthesis, 298, start text, space, K, end text, right parenthesis, end fraction, start text, left parenthesis, p, l, u, g, space, i, n, space, c, o, r, r, e, c, t, space, u, n, i, t, s, space, f, o, r, space, t, h, i, s, space, g, a, s, space, c, o, n, s, t, a, n, t, right parenthesis, end text, n, equals, 0, point, 445, start text, space, m, o, l, e, s, end text, start text, m, o, l, e, c, u, l, e, s, end text, N, equals, 0, point, 445, start text, space, m, o, l, e, s, end text, times, left parenthesis, start fraction, 6, point, 02, times, 10, start superscript, 23, end superscript, start text, space, m, o, l, e, c, u, l, e, s, end text, divided by, 1, start text, space, m, o, l, e, end text, end fraction, right parenthesis, equals, 2, point, 68, times, 10, start superscript, 23, end superscript, start text, space, m, o, l, e, c, u, l, e, s, end text, T, equals, 293, start text, space, K, end text, T, equals, 255, start text, space, K, end text, 255, start text, space, K, end text, point, (startwiththeproportionalversionoftheidealgaslaw), start fraction, P, start subscript, 1, end subscript, V, start subscript, 1, end subscript, divided by, T, start subscript, 1, end subscript, end fraction, equals, start fraction, P, start subscript, 2, end subscript, V, start subscript, 2, end subscript, divided by, T, start subscript, 2, end subscript, end fraction, start text, left parenthesis, s, t, a, r, t, space, w, i, t, h, space, t, h, e, space, p, r, o, p, o, r, t, i, o, n, a, l, space, v, e, r, s, i, o, n, space, o, f, space, t, h, e, space, i, d, e, a, l, space, g, a, s, space, l, a, w, right parenthesis, end text, (volumeisthesamebeforeandaftersincethecanisterisrigid), start fraction, P, start subscript, 1, end subscript, V, divided by, T, start subscript, 1, end subscript, end fraction, equals, start fraction, P, start subscript, 2, end subscript, V, divided by, T, start subscript, 2, end subscript, end fraction, start text, left parenthesis, v, o, l, u, m, e, space, i, s, space, t, h, e, space, s, a, m, e, space, b, e, f, o, r, e, space, a, n, d, space, a, f, t, e, r, space, s, i, n, c, e, space, t, h, e, space, c, a, n, i, s, t, e, r, space, i, s, space, r, i, g, i, d, right parenthesis, end text, start fraction, P, start subscript, 1, end subscript, divided by, T, start subscript, 1, end subscript, end fraction, equals, start fraction, P, start subscript, 2, end subscript, divided by, T, start subscript, 2, end subscript, end fraction, start text, left parenthesis, d, i, v, i, d, e, space, b, o, t, h, space, s, i, d, e, s, space, b, y, space, end text, V, right parenthesis, P, start subscript, 2, end subscript, equals, T, start subscript, 2, end subscript, start fraction, P, start subscript, 1, end subscript, divided by, T, start subscript, 1, end subscript, end fraction, start text, left parenthesis, s, o, l, v, e, space, f, o, r, space, t, h, e, space, p, r, e, s, s, u, r, e, space, end text, P, start subscript, 2, end subscript, right parenthesis, (pluginvaluesforpressureandtemperature, P, start subscript, 2, end subscript, equals, left parenthesis, 255, start text, space, K, end text, right parenthesis, start fraction, 1, start text, space, a, t, m, end text, divided by, 293, start text, space, K, end text, end fraction, start text, left parenthesis, p, l, u, g, space, i, n, space, v, a, l, u, e, s, space, f, o, r, space, p, r, e, s, s, u, r, e, space, a, n, d, space, t, e, m, p, e, r, a, t, u, r, e, end text, right parenthesis, P, start subscript, 2, end subscript, equals, 0, point, 87, start text, space, a, t, m, end text, start text, left parenthesis, c, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, end text, right parenthesis, start text, a, t, m, o, s, p, h, e, r, e, s, end text, P, start subscript, 2, end subscript, equals, 0, point, 87, start text, space, a, t, m, end text, times, left parenthesis, start fraction, 1, point, 013, times, 10, start superscript, 5, end superscript, start text, space, P, a, end text, divided by, 1, start text, space, a, t, m, end text, end fraction, right parenthesis, equals, 88, comma, 200, start text, space, P, a, end text, start text, left parenthesis, c, o, n, v, e, r, t, space, f, r, o, m, space, a, t, m, o, s, p, h, e, r, e, s, space, t, o, space, p, a, s, c, a, l, s, end text, right parenthesis.
