1. Given data

• Molar heat capacity at constant pressure: $C_p = 20.785\ \text{J mol}^{-1}\text{K}^{-1}$
• Temperature change: $T_1=300\ \text{K},\quad T_2=500\ \text{K}$ so $\Delta T = 500-300 = 200\ \text{K}$
• Change in internal energy: $\Delta U = 5000\ \text{J}$
• Gas constant: $R=8.314\ \text{J mol}^{-1}\text{K}^{-1}$

2. Find molar heat capacity at constant volume

For an ideal gas, $C_p - C_v = R$ Hence, $C_v = C_p - R = 20.785 - 8.314 = 12.471\ \text{J mol}^{-1}\text{K}^{-1}$

3. Use the formula for change in internal energy

For an ideal gas, $\Delta U = n C_v \Delta T$ Substitute the values: $5000 = n \times 12.471 \times 200$

4. Solve for number of moles

$n = \frac{5000}{12.471\times 200}$

First compute the denominator: $12.471\times 200 = 2494.2$

So, $n = \frac{5000}{2494.2} \approx 2.0047$

5. Nearest integer

$n \approx 2$

Therefore, the number of moles is: $\boxed{2}$