In a thermodynamic process, pressure of a fixed mass of gas is changed in such manner that the gas releases 30 joule of heat and 18 joule of work was done on the gas. If the initial internal energy of the gas was 60 joule, then the final internal energy is `8x` joule. find the value of x

Options

A. 32 joule
B. 48 joule
C. 72 joule
D. 96 joule

Answer

To solve the problem, we will use the first law of thermodynamics, which states:

\[ \Delta U = Q - W \]

Where:
- \(\Delta U\) is the change in internal energy,
- \(Q\) is the heat added to the system, and
- \(W\) is the work done by the system.

### Step 1: Identify the values given in the problem
- Heat released by the gas, \(Q = -30 \, \text{J}\) (negative because heat is released),
- Work done on the gas, \(W = -18 \, \text{J}\) (negative because work is done on the system),
- Initial internal energy, \(U_i = 60 \, \text{J}\).

### Step 2: Calculate the change in internal energy (\(\Delta U\))
Using the first law of thermodynamics:

\[ \Delta U = Q - W \]

Substituting the values:

\[ \Delta U = (-30) - (-18) \]

\[ \Delta U = -30 + 18 = -12 \, \text{J} \]

### Step 3: Calculate the final internal energy (\(U_f\))
The change in internal energy can also be expressed as:

\[ \Delta U = U_f - U_i \]

Rearranging the equation gives us:

\[ U_f = U_i + \Delta U \]

Substituting the known values:

\[ U_f = 60 + (-12) \]

\[ U_f = 60 - 12 = 48 \, \text{J} \]

### Step 4: Set the final internal energy equal to \(8x\)
According to the problem, the final internal energy is also given as:

\[ U_f = 8x \]

Setting the two expressions for \(U_f\) equal to each other:

\[ 8x = 48 \]

### Step 5: Solve for \(x\)
To find \(x\), divide both sides by 8:

\[ x = \frac{48}{8} = 6 \]

### Final Answer
The value of \(x\) is \(6\).

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