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).
(Given : R = 8.314 JK-1mol-1).
Correct Answer: 2
Approach Solution - 1
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 = nCvΔT, where n is the number of moles, Cv is the molar heat capacity at constant volume, and ΔT is the change in temperature. We know Cp = 20.785 J K−1 mol−1 and R = 8.314 J K−1 mol−1. For an ideal gas, Cp and Cv are related by: Cp = Cv + R. Rearrange this to find Cv:
Cv = Cp − 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 = nCvΔT = 5000 J, solve for n:
n = ΔU / (CvΔT) = 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.
Approach Solution -2
Cp = 20.785 JK-1 mol-1 and ΔU = nCvΔT
∴ nCv = \(\frac{5000}{200}\) = 25
and we know that
Cp – Cv = R
20.785\(-\frac{25}{n} \)= 8.314
n = \(\frac{25}{(20.785-8.314)}\)
= 2
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Concepts Used:
Specific Heat Capacity
Specific heat of a solid or liquid is the amount of heat that raises the temperature of a unit mass of the solid through 1°C.
Molar Specific Heat:
The Molar specific heat of a solid or liquid of a material is the heat that you provide to raise the temperature of one mole of solid or liquid through 1K or 1°C.
Specific Heat at Constant Pressure or Volume:
The volume of solid remains constant when heated through a small range of temperature. This is known as specific heat at a constant volume. It is denoted as CV.
The pressure of solid remains constant when heated through a small range of temperature. This is known as specific heat at constant pressure which can be denoted as CP.