An ideal gas is made to go through a cyclic thermodynamical process in four steps. The amount of heat involved are `Q_1 = 600 J, Q_2 =-400J, Q_3 =-300J` and `Q_4 = 200 J` respectively. The corresponding work involved are `W_1 = 300J, W_2 = -200J, W_3 = -150J` and `W_4`. What is the value of `W_4`

Options

A. -50J
B. 100J
C. 150J
D. 50J

Explanation

To find the value of \( W_4 \) in the cyclic thermodynamic process, we can use the first law of thermodynamics, which states:

\[ \Delta Q = \Delta U + \Delta W \]

For a cyclic process, the change in internal energy \( \Delta U \) is zero because the system returns to its initial state. Therefore, we have:

\[ \Delta Q = \Delta W \]

This means that the total heat added to the system is equal to the total work done by the system over the entire cycle.

### Step 1: Calculate the total heat \( Q_{\text{total}} \)

We sum up the heat involved in each step:

\[ Q_{\text{total}} = Q_1 + Q_2 + Q_3 + Q_4 \]

Substituting the values:

\[ Q_{\text{total}} = 600\, \text{J} + (-400\, \text{J}) + (-300\, \text{J}) + 200\, \text{J} \]

### Step 2: Simplify the total heat calculation

Calculating step by step:

1. \( 600 - 400 = 200 \)
2. \( 200 - 300 = -100 \)
3. \( -100 + 200 = 100 \)

Thus,

\[ Q_{\text{total}} = 100\, \text{J} \]

### Step 3: Calculate the total work \( W_{\text{total}} \)

Now we sum up the work done in each step, including \( W_4 \):

\[ W_{\text{total}} = W_1 + W_2 + W_3 + W_4 \]

Substituting the known values:

\[ W_{\text{total}} = 300\, \text{J} + (-200\, \text{J}) + (-150\, \text{J}) + W_4 \]

### Step 4: Simplify the total work calculation

Calculating step by step:

1. \( 300 - 200 = 100 \)
2. \( 100 - 150 = -50 \)

Thus,

\[ W_{\text{total}} = -50\, \text{J} + W_4 \]

### Step 5: Set the total heat equal to the total work

Since \( Q_{\text{total}} = W_{\text{total}} \), we have:

\[ 100\, \text{J} = -50\, \text{J} + W_4 \]

### Step 6: Solve for \( W_4 \)

Rearranging the equation gives:

\[ W_4 = 100\, \text{J} + 50\, \text{J} = 150\, \text{J} \]

Thus, the value of \( W_4 \) is:

\[ \boxed{150\, \text{J}} \]