First slide

First law of thermodynamics

Question

The P-V diagram of 4 gm of helium gas for a certain process $A \to B$ is shown in the figure. what is the heat given to the gas during the process $A \to B$

Moderate

A

$4 P_{o} V_{o}$
B

$6 P_{o} V_{o}$
C

$4.5 P_{o} V_{o}$
D

$2 P_{o} V_{o}$

Solution

Change in internal energy from $A \to B$ is
$\begin{array}{l} Δ U = \frac{f}{2} μ R Δ T = \frac{f}{2} (P_{f} V_{f} - P_{i} V_{i}) \\ = \frac{3}{2} (2 P_{0} \times 2 V_{0} - P_{0} \times V_{0}) = \frac{9}{2} P_{0} V_{0} \end{array}$
Work done in process $A \to B$ is equal to the Area covered by the graph with volume axis i.e.,
$W_{A \to B} = \frac{1}{2} (P_{0} + 2 P_{0}) \times (2 V_{0} - V_{0}) = \frac{3}{2} P_{0} V_{0}$
$H e n c e, Δ Q = Δ U + Δ W = \frac{9}{2} P_{0} V_{0} + \frac{3}{2} P_{0} V_{0} = 6 P_{0} V_{0}$

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