At constant volume, 4 mol of an ideal gas when heated from 300K to 500K changes its internal energy by 5000J. The molar heat capacity at constant volume is

We are dealing with an ideal gas that is heated at constant volume, so we recall the basic thermodynamic relation for an ideal gas kept at constant volume:

$$\Delta U \;=\; n\,C_{v,m}\,\Delta T$$

where

$$\Delta U = \text{change in internal energy},$$
$$n = \text{number of moles},$$
$$C_{v,m} = \text{molar heat capacity at constant volume},$$
$$\Delta T = \text{change in temperature (}T_{\text{final}} - T_{\text{initial}}\text{)}.$$

From the data given in the question we have

$$n = 4\ \text{mol},$$
$$T_{\text{initial}} = 300\ \text{K},$$
$$T_{\text{final}} = 500\ \text{K},$$
$$\Delta U = 5000\ \text{J}.$$

First we find the temperature change:

$$\Delta T = T_{\text{final}} - T_{\text{initial}} = 500\ \text{K} - 300\ \text{K} = 200\ \text{K}.$$

Now we substitute every known value into the formula $$\Delta U = n\,C_{v,m}\,\Delta T$$:

$$5000\ \text{J} \;=\; (4\ \text{mol})\,C_{v,m}\,(200\ \text{K}).$$

We isolate $$C_{v,m}$$ by dividing both sides by the product $$n\Delta T$$:

$$C_{v,m} \;=\;\frac{\Delta U}{n\,\Delta T} \;=\;\frac{5000\ \text{J}}{4\ \text{mol}\times 200\ \text{K}}.$$

Carrying out the multiplication in the denominator first:

$$4\ \text{mol}\times 200\ \text{K} = 800\ \text{mol·K}.$$

So we have

$$C_{v,m} \;=\;\frac{5000\ \text{J}}{800\ \text{mol·K}}.$$

Finally, performing the division:

$$C_{v,m} \;=\; 6.25\ \text{J mol}^{-1}\text{K}^{-1}.$$

So, the answer is $$6.25\ \text{J mol}^{-1}\text{K}^{-1}.$$