The percentage change in the pressure of gas at constant temperature is 200% (increases). Calculate the percentage change in the volume of the gas.

Answer

To solve the problem, we need to find the percentage change in the volume of a gas when the pressure increases by 200% at constant temperature. We will use the ideal gas law and the relationship between pressure and volume.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The percentage change in pressure is given as 200%. This means:
\[ \frac{P_2 - P_1}{P_1} \times 100 = 200\% \]

2. **Convert Percentage Change to Pressure Values:**
- From the percentage change, we can express it in terms of pressures:
\[ P_2 - P_1 = 2P_1 \]
- Rearranging gives:
\[ P_2 = P_1 + 2P_1 = 3P_1 \]

3. **Use the Ideal Gas Law:**
- According to the ideal gas law, at constant temperature (T), the product of pressure (P) and volume (V) is constant:
\[ P_1 V_1 = P_2 V_2 \]

4. **Substitute the Values:**
- Substitute \(P_2\) into the equation:
\[ P_1 V_1 = (3P_1) V_2 \]
- We can cancel \(P_1\) from both sides (assuming \(P_1 \neq 0\)):
\[ V_1 = 3 V_2 \]

5. **Express \(V_2\) in Terms of \(V_1\):**
- Rearranging gives:
\[ V_2 = \frac{V_1}{3} \]

6. **Calculate the Change in Volume:**
- The change in volume can be calculated as:
\[ \text{Change in Volume} = V_2 - V_1 = \frac{V_1}{3} - V_1 = \frac{V_1 - 3V_1}{3} = \frac{-2V_1}{3} \]

7. **Calculate the Percentage Change in Volume:**
- The percentage change in volume is given by:
\[ \text{Percentage Change} = \frac{V_2 - V_1}{V_1} \times 100 = \frac{\frac{-2V_1}{3}}{V_1} \times 100 \]
- Simplifying this gives:
\[ = \frac{-2}{3} \times 100 = -66.67\% \]

### Final Answer:
The percentage change in the volume of the gas is approximately **-66.67%**. The negative sign indicates that the volume has decreased.