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 $8\text{g}$ of oxygen, if it is heated from $200\text{K}$ to $300\text{K}$ at constant volume? Assume oxygen to be an ideal gas. (Given, $a=1.2{\text{JK}}^{-1}{\text{mol}}^{-1}$, $b=12.8\times {10}^{-2}{\text{JK}}^{-2}{\text{mol}}^{-1}$, $c=1.0\times {10}^{-7}{\text{JK}}^{-3}{\text{mol}}^{-1}$).

• A. $850.60\text{J}$
• B. $950.50\text{J}$
• C. $830.16\text{J}$
• D. $853.62\text{J}$

Answer 1

The correct option is C $830.16\text{J}$

Given,
$C_v=a+bT+c{T}^2$
We know, $dE=n{C}_{v}dT\text{Where E= internal energy}$
Therefore change in internal energy,
${\int }_{E_1}^{E_2}dE=n{\int }_{T_1}^{T_2}(a+bT+c{T}^2)dT$
$\Rightarrow \Delta E=E_2-E_1=n{[aT+\frac{b{T}^2}{2}+\frac{c{T}^3}{3}]}_{200}^{300}$
$\Rightarrow \Delta E=n\left[a\right(300-200)+\frac{b}{2}[(300{)}^2-(200{)}^2]+\frac{c}{3}[(300{)}^3-(200{)}^3\left]\text{J}\right]$
$\Rightarrow \Delta E=\frac{8}{32}[100a+2.5\times {10}^{4}b+6.33\times {10}^{6}c\text{J}]$
$\Rightarrow \Delta E=\frac{1}{4}[100\times 1.2+2.5\times {10}^4\times 12.8\times {10}^{-2}+6.33\times {10}^6\times {10}^{-7}]\text{J}$
$\Rightarrow \Delta E=\frac{100}{4}[1.2+32+0.00633]\text{J}$
$\Rightarrow \Delta E=830.158\text{J}\approx 830.16\text{J}$