Question

Find the change in internal energy at constant volume if one mole of gas is heated from $10{}^\circ C$ to $30{}^\circ C$. Mentioned that the specific heat at constant volume is, ${{C}_{V}} = 24Jmo{{l}^{-1}}{{K}^{-1}}$

Answer 1

Hint: As the change in volume is mentioned as zero, the work done will be zero. In accordance with the first law of thermodynamics, it gives a result which tells that the internal energy will be equivalent to the change in heat. The change in internal energy at constant volume is explained as the product of the number of moles, specific heat capacity at constant volume and the change in temperature. These all may help you in solving this question.

Complete step-by-step answer:
First of all let us mention what all are given in the question. The specific heat at constant volume is given as,
${{C}_{V}}=24Jmo{{l}^{-1}}{{K}^{-1}}$
The number of moles is given as,
$n=1$
Change in temperature is given as,
$\Delta T=30-10=20{}^\circ C$
This should be converted into kelvin. As we know here the change in temperature is calculated. As the change in temperature in degree celsius and in kelvin are similar, the value change in temperature will be the same,
That is we can write that,
$\Delta T=30{}^\circ C-10{}^\circ C=303K-283K=20K$
Therefore, now we can calculate the internal energy.
According to the first law of thermodynamics,
$\Delta U=\Delta q+\Delta W$
In which the change in work done is given by the equation,
$\Delta W=P\Delta V$
As the volume is constant here, the change in volume will be zero. Thus the work done will also be zero. Therefore we can write that,
$\begin{align}
  & \Delta U=\Delta q+0 \\
 & \Delta U=\Delta q \\
\end{align}$
The change in internal energy will be equivalent to the change in heat energy.
That is,
$\Delta U=\Delta q=n{{C}_{V}}\Delta T$
Let us substitute the values in it,
$\Delta U=1\times 24\times 20=480J$
Therefore the correct answer is obtained.

Note: Enthalpy is the measure of the amount of heat content used or released in a system at a fixed pressure. The measure of heat content given off or absorbed when a reaction is taking place at constant volume is equivalent to the change in the internal energy of the system. In a cyclic system, the change in internal energy is zero.