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Standard XII

Physics

Gas Pressure and Temperature

Methane consi...

Question

Methane (considered to be an ideal gas) initially at $25^{o}$ C and 1 bar pressure is heated at constant pressure until the volume has doubled. The variation of the molar heat capacity with absolute temperature is given by: $C_{P} = 22.34 + 48.1 \times 10^{- 3} T$ . where $C_{P}$ is in $J K^{- 1} m o l^{- 1}$ . Calculate molar (a) $Δ H$ (b) $Δ U$

(a)

13.064 k J m o l^{- 1}

(b)

10.587 k J m o l^{- 1}

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(a)

130.64 k J m o l^{- 1}

(b)

105.87 k J m o l^{- 1}

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(a)

1306.4 k J m o l^{- 1}

(b)

- 10.587 k J m o l^{- 1}

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None of these

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Solution

The correct option is A (a) $13.064 k J m o l^{- 1}$ (b) $10.587 k J m o l^{- 1}$
Given,

$T_{1} = 25^{o} C, P = 1 b a r$

Ideal gas equation,

$P V = n R T$
Since, pressure is constant-

$\frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}}$

$V_{2} = 2 V_{1}$ $\Rightarrow T_{2} = 2 T_{1} = 50^{o} C = 323 K$

$C_{P} = 22.34 + 48.1 \times 10^{3} T$

$Δ H = \int^{n = 1} n C_{P} d T = \int_{T_{1}}^{T_{2}} (22.34 + 48.1 \times 10^{- 3} T) d T = 22.34 \times 9298 + \frac{48.1 \times 10^{- 3} \times 298 \times 3}{2} \times 298$

(a) $Δ H = 13.064 k J / m o l e$

$W = P (V_{2} - V_{1}) = n R (T_{2} - T_{1})$

$W = 2.477 K J$

$Δ U = (13.064 - 2.477) K J$

(b) $Δ U = 10.587 k J$

Option (A) is correct.

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