Question

The molar heat capacity for an ideal gas at constant pressure is 20.785 J K^–1 mol^–1. The change in internal energy is 5000 J upon heating it from 300 K to 500 K. The number of moles of the gas at constant volume is _____. (Nearest integer)
(Given : R = 8.314 JK-1mol-1).

Answer 1

Correct Answer: 2

To find the number of moles of the gas, we use the relation for the change in internal energy (ΔU) at constant volume for an ideal gas: ΔU = nC_vΔT, where n is the number of moles, C_v is the molar heat capacity at constant volume, and ΔT is the change in temperature. We know C_p = 20.785 J K⁻¹ mol⁻¹ and R = 8.314 J K⁻¹ mol⁻¹. For an ideal gas, C_p and C_v are related by: C_p = C_v + R. Rearrange this to find C_v:

C_v = C_p − R = 20.785 − 8.314 = 12.471 J K⁻¹ mol⁻¹. 

Now, calculate ΔT: ΔT = 500 K − 300 K = 200 K. Using the change in internal energy: ΔU = nC_vΔT = 5000 J, solve for n:

n = ΔU / (C_vΔT) = 5000 / (12.471 × 200).

Calculate:
n ≈ 5000 / 2494.2 ≈ 2.004.

The nearest integer value for n is 2. This value is within the given range (2,2). Therefore, the number of moles of the gas at constant volume is 2.

Answer 2

C_p = 20.785 JK^-1 mol^-1 and ΔU = nC_vΔT
∴ nC_v = \(\frac{5000}{200}\) = 25
and we know that
C_p – C_v = R
20.785\(-\frac{25}{n} \)= 8.314
n = \(\frac{25}{(20.785-8.314)}\)
= 2