Question 3.6: One mole of gas undergoes a change from an initial state des...
One mole of N_{2} gas undergoes a change from an initial state described by T =200. k and P_{i}=5.00 bar to a final state described by T =400.K and P_{f}=20.0 bar. treat N_{2} as a van der Waals gas with the parameters \alpha= 0.137 Pa m^{6} mol^{-2} and b=3.87 ×10^{-5} m^{3} mol^{-1}. We use the path N_{2}\left( g,T =200.K,P=5.00 bar \right)\to N_{2}\left( g,T =200.K,P=20.0 bar \right)\to N_{2}\left( g,T =400.K,P=20.0 bar \right), keeping in mind that all paths will give the same answer for \Delta U of the overall process. a. Calculate \Delta U_{T}=\int_{V_{i}}^{V_{f}}\left( \partial U/\partial V \right)_{T}dV using the result of Example Problem 3.5. Note that V_{i} = 3.28 × 10^{-3} m^{3} and V_{f} = 7.88 × 10^{-4} m^{3} at 200. K, as calculated using the van der Waals equation of state. b. Calculate \Delta U_{V}=n\int_{V_{i}}^{V_{f}}C_{V,m}dT using the following relationship for C_{V,m} in this temperature range:
\frac{C_{V,m}}{J K^{-1} mol^{-1}} =22.50 -1.187 ×10^{-2}\frac{T}{K}+2.3968 ×10^{-5}\frac{T^{2}}{K^{2}}-1.0176 ×10^{-8}\frac{T^{3}}{K^{3}}
The ratios T^{n}/K^{n} ensure that C_{V,m} has the correct units.
c. Compare the two contributions to \Delta U .can \Delta U_{T} be neglected relative to \Delta U_{V} ?
a. Using the result of Example Problem 3.5,
\Delta U_{T}=n^{2}\alpha\left( \frac{1}{V_{m,i}}-\frac{1}{{V_{m,f}}}\right)=0.137 Pa m^{6}×\left( \frac{1}{3.28 × 10^{-3}m^{3}} -\frac{1}{7.88× 10^{-4}m^{3}}\right)=-132 J
b. \Delta U_{V}=n\int_{T_{i}}^{T_{f}}C_{V,m}dT
=\int_{200.}^{400.}\left( ^{22.50-1.187 ×10^{-2}\frac{T}{K}+2.3968 × 10^{-5}\frac{T^{2}}{K^{2}}}_{-1.0176 × 10^{-8}\frac{T^{3}}{K^{3}}} \right)d\left( \frac{T}{K} \right)J= \left( 4.50 -0.712 +0.447 -0.0610 \right)KJ =4.17 KJ
c. \Delta U_{T} is 3.2\% of \Delta U_{V} for this case. In this example, and for most processes, \Delta U_{T} can be neglected relative to \Delta U_{V} for real gases.