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To solve the problem involving the ammonia gas (NH₃), we can use the **Ideal Gas Law**:

\[
PV = nRT
\]

Since the number of moles (n) and the gas constant (R) are constants, we can simplify this to:

\[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
\]

Where:
- \(P_1\), \(V_1\), \(T_1\) are the initial pressure, volume, and temperature.
- \(P_2\), \(V_2\), \(T_2\) are the final pressure, volume, and temperature.

We are given:
- The initial temperature \(T_1 = 13.0^\circ C = 286.15\,K\).
- The final temperature \(T_2 = -1.0^\circ C = 272.15\,K\).
- The pressure is increased by 15%, meaning \(P_2 = 1.15 \times P_1\).

### Step 1: Relationship of Volume
We need to determine if the volume increases, decreases, or stays the same. By rearranging the equation for volume, we get:

\[
\frac{V_2}{V_1} = \frac{P_1 T_2}{P_2 T_1}
\]

Substitute the known values:

\[
\frac{V_2}{V_1} = \frac{1 \times 272.15}{1.15 \times 286.15} = \frac{272.15}{328.07} \approx 0.83
\]

Since \(\frac{V_2}{V_1} < 1\), the volume decreases.

### Step 2: Percentage Change in Volume
To calculate the percentage change:

\[
\text{Percentage change} = \left(1 - \frac{V_2}{V_1}\right) \times 100\% = \left(1 - 0.83\right) \times 100\% = 17\%
\]

Thus, the volume decreases by approximately 17%.

---

Would you like any further details or clarifications?

Here are 5 related questions to explore:
1. How does the Ideal Gas Law relate pressure, volume, and temperature?
2. What assumptions are made when treating ammonia as an ideal gas?
3. How do pressure changes affect the volume of a gas?
4. What would happen if the temperature were lowered further below -1°C?
5. How do real gases differ from ideal gases at low temperatures?

**Tip**: When working with gas laws, always convert temperatures to Kelvin by adding 273.15 to the Celsius value!

Suppose the temperature of a sample of ammonia gas is lowered from 13.0 °C to −1.0 °C, and at the same time the pressure is increased by 15.0%. Does the volume of the sample increase, decrease, or stay the same? If you said the volume increases or decreases, calculate the percentage change in the volume.

Learn how to calculate the volume change in ammonia gas when temperature decreases and pressure increases, using the Ideal Gas Law. This problem involves understanding how pressure, temperature, and volume are related. Suitable for Grades 10-12.

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