Forty calories of heat is needed to raise the temperature of 1mole of an ideal monatomic gas from 20∘C to 30∘C at a constant pressure. The amount of heat required to raise its temperature over the same interval at a constant volume (R=2calmol⁻¹K⁻¹) is:

The correct option is 20 calorie.

Step-by-Step Explanation:
We are given that for $n=1$ mole of an ideal monatomic gas, the heat required to raise its temperature by $\Delta T=30{\circ }^{‘}C-20{\circ }^{‘}C=10K$ at a constant pressure ($Q_p$) is 40 calories.
The gas constant is given as $R=2{\text{cal mol}}^{-1}{\text{K}}^{-1}$.

For a process at constant pressure, the heat supplied is given by the formula:
$Q_p=n{C}_p\Delta T$
where $C_p$ is the molar heat capacity at constant pressure.

Substituting the given values:
$40=1\times C_p\times 10$
Solving for $C_p$:
$C_p=\frac{40}{10}=4{\text{cal mol}}^{-1}{\text{K}}^{-1}$

According to Mayer's relation for an ideal gas:
$C_p-C_v=R$
where $C_v$ is the molar heat capacity at constant volume.

Substituting the values of $C_p$ and $R$:
$4-C_v=2$
$C_v=4-2=2{\text{cal mol}}^{-1}{\text{K}}^{-1}$

Now, we calculate the amount of heat required to raise the temperature of the same gas over the same temperature interval ($\Delta T=10K$) at a constant volume ($Q_v$):
$Q_v=n{C}_v\Delta T$
Substituting the values:
$Q_v=1\times 2\times 10=20\text{calories}$

Thus, the amount of heat required at constant volume is 20 calories.