The volume of a certain quantity of oxygen is `200 cm^(3)` at `15^(@)C` and `0.98` bar pressure of Hg. What should be the temperature so that the volume becomes `180 cm^(3)` under a pressure of `1 "bar"` of Hg?

Explanation

To solve the problem, we will use the combined gas law, which relates the pressure, volume, and temperature of a gas. The formula is given by:

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

Where:
- \( P_1 \) = initial pressure
- \( V_1 \) = initial volume
- \( T_1 \) = initial temperature (in Kelvin)
- \( P_2 \) = final pressure
- \( V_2 \) = final volume
- \( T_2 \) = final temperature (in Kelvin)

### Step 1: Convert the initial temperature from Celsius to Kelvin
The initial temperature \( T_1 \) is given as \( 15^\circ C \). To convert Celsius to Kelvin, we use the formula:

\[ T(K) = T(°C) + 273 \]

So,

\[ T_1 = 15 + 273 = 288 \, K \]

### Step 2: Identify the known values
From the problem, we have:
- \( V_1 = 200 \, cm^3 \)
- \( V_2 = 180 \, cm^3 \)
- \( P_1 = 0.98 \, bar \)
- \( P_2 = 1 \, bar \)
- \( T_1 = 288 \, K \)

### Step 3: Rearrange the combined gas law to solve for \( T_2 \)
We want to find \( T_2 \), so we rearrange the equation:

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

### Step 4: Substitute the known values into the equation
Now we substitute the values into the rearranged equation:

\[ T_2 = \frac{1 \, bar \times 180 \, cm^3 \times 288 \, K}{0.98 \, bar \times 200 \, cm^3} \]

### Step 5: Calculate \( T_2 \)
Calculating the numerator:

\[ 1 \times 180 \times 288 = 51840 \]

Calculating the denominator:

\[ 0.98 \times 200 = 196 \]

Now, substituting these values into the equation for \( T_2 \):

\[ T_2 = \frac{51840}{196} \approx 264.49 \, K \]

### Step 6: Convert \( T_2 \) from Kelvin to Celsius
To convert Kelvin back to Celsius, we use the formula:

\[ T(°C) = T(K) - 273 \]

So,

\[ T_2 = 264.49 - 273 \approx -8.51 \, °C \]

### Final Answer
The temperature \( T_2 \) required for the volume to become \( 180 \, cm^3 \) under \( 1 \, bar \) of pressure is approximately \( -8.51 \, °C \).

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