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One mole of an ideal monoatomic gas requires 207 J heat to raise the temperature by 10 K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same 10 K, the heat required is [Given the gas constant R = 8.3 J/ mol. K] [1990]
  • a)
    198.7 J
  • b)
    29 J
  • c)
    215.3 J
  • d)
    124 J
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
One mole of an ideal monoatomic gas requires 207 J heat to raise the t...
We know that  Qv = nCvΔT and Qp = nCpΔT
Given Qp = 207 J
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One mole of an ideal monoatomic gas requires 207 J heat to raise the t...
Given:
- Heat required to raise the temperature of 1 mole of an ideal monoatomic gas by 10 K at constant pressure = 207 J
- Gas constant, R = 8.3 J/mol.K

To find:
- Heat required to raise the temperature of the same gas by 10 K at constant volume

Solution:

1. Heat capacity at constant pressure (Cp) can be calculated as:

Cp = (ΔH / ΔT)p

where, ΔH = change in enthalpy, ΔT = change in temperature, and p = constant pressure

2. Heat capacity at constant volume (Cv) can be calculated as:

Cv = (ΔU / ΔT)v

where, ΔU = change in internal energy, and v = constant volume

For an ideal gas, internal energy depends only on temperature, so

ΔU = nCvΔT

where, n = number of moles of gas

3. The relationship between Cp and Cv for an ideal gas is:

Cp - Cv = nR

where, R = gas constant

Using these equations, we can find the heat required to raise the temperature of the gas at constant volume:

ΔU = nCvΔT

ΔH = ΔU + pΔV

where, ΔV = change in volume

At constant volume, ΔV = 0, so ΔH = ΔU

Cv = (ΔU / ΔT)v

ΔU = nCvΔT

Substituting these values,

Cv = (ΔU / ΔT)v = (ΔH / ΔT)v - pΔV / ΔT

Cp - Cv = nR

Cp = Cv + nR

ΔH = ΔU = nCvΔT

ΔH = nCpΔT - nRΔT

207 J = nCp(10 K) - nR(10 K)

nCp - nR = 20.7 J/mol

Cv = Cp - nR

Cv = 20.7 J/mol / n

For 1 mole of gas,

Cv = 20.7 J/mol

ΔU = nCvΔT = (1 mol)(20.7 J/mol)(10 K) = 207 J

Therefore, the heat required to raise the temperature of the gas by 10 K at constant volume is 207 J.

Answer: Option D (124 J) is incorrect.
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One mole of an ideal monoatomic gas requires 207 J heat to raise the temperature by 10 K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same 10 K, the heat required is [Given the gas constant R = 8.3 J/ mol. K] [1990]a)198.7 Jb)29 Jc)215.3 Jd)124 JCorrect answer is option 'D'. Can you explain this answer?
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