At what pressure in atm does ethane (C₂H₆) have a density of 37.2 g/L at 73.0 °C?

To find the pressure at which ethane (C₂H₆) has a density of 37.2 g/L at 73.0 °C, we can use the **Ideal Gas Law**:

\[
PV = nRT
\]

Where:
- \( P \) is the pressure in atm (which we need to find),
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the ideal gas constant (\(0.0821 \, \text{L·atm/mol·K}\)),
- \( T \) is the temperature in Kelvin.

We will also use the equation that relates mass, molar mass, and volume to density:

\[
\text{Density} = \frac{\text{mass}}{\text{volume}} = \frac{n \cdot M}{V}
\]

Where:
- \( \text{Density} = 37.2 \, \text{g/L} \),
- \( M \) is the molar mass of ethane, which is \( 30.07 \, \text{g/mol} \) (for \( C_2H_6 \)),
- \( n \) is the number of moles,
- \( V \) is the volume.

### Step 1: Convert the temperature to Kelvin
\[
T = 73.0 \, ^\circ\text{C} + 273.15 = 346.15 \, \text{K}
\]

### Step 2: Rearrange the Ideal Gas Law
From the Ideal Gas Law, we have \( P = \frac{nRT}{V} \), and using the relationship for density:

\[
P = \frac{\text{Density} \cdot R \cdot T}{M}
\]

### Step 3: Plug in the known values
\[
P = \frac{(37.2 \, \text{g/L}) \cdot (0.0821 \, \text{L·atm/mol·K}) \cdot (346.15 \, \text{K})}{30.07 \, \text{g/mol}}
\]

### Step 4: Calculate the pressure
\[
P = \frac{(37.2) \cdot (0.0821) \cdot (346.15)}{30.07} = \frac{1053.2}{30.07} \approx 35.03 \, \text{atm}
\]

Thus, the pressure is approximately **35.03 atm**.

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Would you like more details or have any questions? 

Here are 5 related questions you might explore:
1. How does changing temperature affect the pressure in the ideal gas law?
2. What assumptions does the ideal gas law make about gas behavior?
3. How does the molar mass of a gas influence its density?
4. What would the pressure be if the temperature were reduced to 25°C?
5. How would the calculation change if the gas was not ideal?

**Tip:** Always check the units in your calculations to ensure consistency, especially when using the ideal gas constant \(R\).

Find the pressure of ethane (C₂H₆) with a density of 37.2 g/L at 73.0 °C using the Ideal Gas Law. This solution demonstrates key concepts such as density, gas laws, and temperature conversion. Suitable for high school students in Grades 10-12.

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