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A gas is compressed at a constant pressure of $50N{m^{ - 2}}$ from a volume $10{m^3}$ to a volume of $4{m^3}$. If $100J$ of heat is added to the gas, then its internal energy
A. increases by $400J$
B. increases by $200J$
C. decreases by $400J$
D. decreases by $200J$

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Last updated date: 19th Jun 2024
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Answer
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Hint: This problem can be solved simply by applying the concept of the first law of thermodynamics. Applying the above values in the first law equation, we can obtain the value of the change in internal energy.

Formulas used
$dQ = dU + dW$ where $dQ$ is the heat absorbed by the system, $dU$ is the increase in the internal energy and $dW$ is the external work done.
$dW = PdV$ where $P$ is the pressure and $dV$ is change in volume of the system.

Complete step by step answer
According to the first law of thermodynamics, when heat energy is supplied to a system, a part of it is used to increase the internal energy of the system and the rest of it is used to perform external work.
The statement for first law of thermodynamics is given as,
$dQ = dU + dW$ where $dQ$ is the heat absorbed by the system, $dU$ is the increase in the internal energy and $dW$ is the external work done.
Now, $dW$ can be written as $PdV$ where $P$ is the pressure and $dV$ is change in volume of the system.
Thus, the above equation becomes,
$dQ = dU + PdV$
$ \Rightarrow dU = dQ - PdV$
The volume of the gas given changes from $10{m^3}$ to $4{m^3}$
So, ${V_1} = 10{m^3}$ and ${V_2} = 4{m^3}$
$\begin{gathered}
  dV = \left( {{V_2} - {V_1}} \right) = \left( {4 - 10} \right){m^3} \\
   \Rightarrow dV = - 6{m^3} \\
\end{gathered} $
Therefore,
$\begin{gathered}
  dU = 100 - \left\{ {50 \times \left( { - 6} \right)} \right\}J \\
   \Rightarrow dU = 100 + 300 \\
   \Rightarrow dU = 400J \\
\end{gathered} $
This means that the internal energy of the gas increases by $400J$

Thus, the correct option is A.

Note: The first law establishes an exact relation between heat and other forms of energy. However, it cannot state the condition under which a system can transform its heat energy into work. Also cannot specify how much of the heat energy can be converted to work. The internal energy of a system is a state function, which means that it only depends upon the initial and final state and is independent of the path.