It is found that with an increase in temperature by `40%`, the volume decreases by `20%` with change in pressure. Find the percentage change in pressure.

Options

A. `40%` decrease
B. `60%` decrease
C. `75%` increase
D. `80%` increase

Answer

To solve the problem, we will use the ideal gas law and the relationships between pressure, volume, and temperature. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- Initial temperature \( T \)
- Final temperature \( T' = T + 40\% \text{ of } T = 1.4T \)
- Initial volume \( V \)
- Final volume \( V' = V - 20\% \text{ of } V = 0.8V \)

### Step 2: Use the Ideal Gas Law
According to the ideal gas law, we have:
\[ PV = nRT \]
This can be rearranged to show the relationship between pressure, volume, and temperature:
\[ \frac{PV}{T} = \text{constant} \]
Thus, for initial conditions and final conditions, we can write:
\[ \frac{PV}{T} = \frac{P'V'}{T'} \]

### Step 3: Substitute the Values
Substituting the known values into the equation:
\[ \frac{P \cdot V}{T} = \frac{P' \cdot (0.8V)}{(1.4T)} \]

### Step 4: Simplify the Equation
Cancel \( V \) from both sides:
\[ \frac{P}{T} = \frac{P' \cdot 0.8}{1.4} \]
Rearranging gives:
\[ P' = \frac{1.4P}{0.8} \]

### Step 5: Calculate \( P' \)
Now, simplify the expression for \( P' \):
\[ P' = 1.75P \]

### Step 6: Find the Percentage Change in Pressure
The percentage change in pressure is given by:
\[ \text{Percentage Change} = \frac{P' - P}{P} \times 100 \]
Substituting \( P' \):
\[ \text{Percentage Change} = \frac{1.75P - P}{P} \times 100 = \frac{0.75P}{P} \times 100 = 75\% \]

### Conclusion
The percentage change in pressure is an increase of **75%**.

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