Question

The molar heat capacity of oxygen gas is given by the expression $C_v=a+bT+c{T}^2$ where a, b and c are constants. What will be the change in the internal energy of 8 g of oxygen, if it is heated from 200 K to 300 K at constant volume? Assume oxygen to be an ideal gas. (Given, $a=1.2J{K}^{-1}mo{l}^{-1}$, $b=12.8\times {10}^{-2}J{K}^{-2}mo{l}^{-1}$, $c=1.0\times {10}^{-7}J{K}^{-3}mo{l}^{-1}$).

• A. $830.16J$
• B. $853.62J$
• C. $950.50J$
• D. $850.60J$

Answer 1

The correct option is A $830.16J$

Given,
$C_v=a+bT+c{T}^2$
We know, $dE=n{C}_{v}dTWhereE=internalenergy$
Therefore change in internal energy,
${\int }_{E_1}^{E_2}dE=n{\int }_{T_1}^{T_2}(a+bT+c{T}^2)dT$
$\Delta E=E_2-E_1=n{[T+\frac{b{T}^2}{2}+\frac{c{T}^3}{3}]}_{200}^{300}$
$\Rightarrow \Delta E=n\left[a\right(300-200)+\frac{b}{2}[(300{)}^2-(200{)}^2]+\frac{c}{3}[(300{)}^3-(200{)}^3\left]\right]$
$\Rightarrow \Delta E=\frac{8}{32}[100a+2.5\times {10}^{4}b+6.3\times {10}^{6}c]$
$\Rightarrow \Delta E=\frac{1}{4}[100\times 1.2+2.5\times {10}^4\times 12.8\times {10}^{-2}+6.3\times {10}^6\times {10}^{-7}]$
$\Rightarrow \Delta E=830.157$