When $1$ mole of gas is heated at constant volume, the temperature is raised from $298$ to $308K$ and heat supplied to gas is $500J$. Which of the following statements is true?

The correct option is C

$\mathrm{q}=∆\mathrm{U}=500\mathrm{J}$,$\mathrm{W}=0$

Step 1: Given data

Heat supplied, $\mathrm{q}=500\mathrm{J}$

The temperature of the system is raised from $298$ to $308K$

For this process, volume is constant i.e. $∆\mathrm{V}=0$

Step 2: Formula used

Work done is given by,

$\begin{array}{rcl}& \Rightarrow & \mathrm{W}=-\mathrm{P}∆\mathrm{V}\\ & =& -\mathrm{P}\times 0\\ & =& 0\end{array}$

Step 3: Calculation

Now applying the first law of thermodynamics and using $W=0$, we got :

$$\begin{gathered} \Rightarrow ∆\mathrm{U}=\mathrm{q}+\mathrm{W} \\ =\mathrm{q}+0 \\ =\mathrm{q} \end{gathered}$$

So we got, $\mathrm{q}=∆\mathrm{U}=500\mathrm{J}$ and $\mathrm{W}=0$.