< Ideal gas lawRevision as of 08:00, 7 January 2009 by imported>Paul Wormer
- All gases mentioned below are assumed to be ideal, i.e. their p, V, T dependence is given by the ideal gas law.
- The molar gas constant R = 0.082057 atm⋅L/(K⋅mol)
Example problems
Problem 1
Determine the volume of 1 mol of ideal gas at pressure 1 atm and temperature 20 °C.
![{\displaystyle V={\frac {n\,R\,T}{p}}={\frac {1\cdot 0.082057\cdot (20+273.15)}{1}}\quad {\frac {\mathrm {mol} \cdot {\frac {\mathrm {atm} \cdot \mathrm {L} }{\mathrm {K} \cdot \mathrm {mol} }}\cdot \mathrm {K} }{\mathrm {atm} }}=24.0550\quad \mathrm {L} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/330ff4082ac52351f213c438f70c01dcead35157)
Problem 2
Compute from Charles' and Gay-Lussac's law (V/T is constant) the volume of an ideal gas at 1 atm and 0 °C (Use the final result of the previous problem). Write VT for the volume at T °C, then
![{\displaystyle {\frac {V_{20}}{273.15+20}}={\frac {V_{0}}{273.15+0}}\quad \Longrightarrow V_{0}=273.15\times {\frac {24.0550}{298.15}}=22.4139\;\mathrm {L} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bd3de97829aa9b2bb2b96ebf6408111c4bf13c)
Problem 3
A certain amount of gas that has an initial pressure of 1 atm and an initial volume of 2 L, is compressed to a final pressure of 5 atm at constant temperature. What is the final volume of the gas?
Boyle's law (pV is constant)
![{\displaystyle (1.1)\qquad \qquad p_{\mathrm {i} }\,V_{\mathrm {i} }=p_{\mathrm {f} }\,V_{\mathrm {f} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99fb7b183ba51179627f807643fb52342b84712f)
or
![{\displaystyle (1.2)\qquad \qquad V_{\mathrm {f} }={\frac {p_{\mathrm {i} }\;V_{\mathrm {i} }}{p_{\mathrm {f} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eff50c22a09ebeef9575c0f69bfb9bdbda4d52)
Inserting the given numbers
![{\displaystyle (1.3)\qquad \qquad V_{\mathrm {f} }=\left({\frac {1\cdot 2}{5}}\right)\;{\frac {\mathrm {atm} \cdot \mathrm {L} }{\mathrm {atm} }}=0.4\;\mathrm {L} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb916dd1aaf495f7cc669f4acb3fe80da9936aa)
Ideal gas law
The number n of moles is constant
![{\displaystyle (1.4)\qquad \qquad pV=nRT\quad \Longrightarrow \quad n={\frac {p_{\mathrm {i} }\,V_{\mathrm {i} }}{RT_{\mathrm {i} }}}={\frac {p_{\mathrm {f} }\,V_{\mathrm {f} }}{RT_{\mathrm {f} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e735d4f7dde0f6ca2b04f362c1798d4948d2940c)
It is given that the initial and final temperature are equal,
, therefore the products RT on both sides of the equation cancel, and Eq. (1.4) reduces to Eq. (1.1).
Problem 4
How many moles of nitrogen are present in a 50 L tank at 25 °C when the pressure is 10 atm? Numbers include only 3 significant figures.
![{\displaystyle n={\frac {p\,V}{R\,T}}={\frac {10.0\cdot 50.0}{0.0821\cdot (273+25.0)}}\quad {\frac {\mathrm {atm} \cdot \mathrm {L} }{{\frac {\mathrm {atm} \cdot \mathrm {L} }{\mathrm {K} \cdot \mathrm {mol} }}\cdot \mathrm {K} }}={\frac {500}{0.0821\cdot 298}}\quad {\frac {\mathrm {mol} \cdot \mathrm {atm} \cdot \mathrm {L} }{\mathrm {atm} \cdot \mathrm {L} }}=20.4\quad \mathrm {mol} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1be35492bc5eaff2c9c5ffe9568a1cb12f9fbe3e)