Question

Calculate the final Celsius temperature of nitrogen dioxide gas if \(1.95 \mathrm{~L}\) of the gas at \(0^{\circ} \mathrm{C}\) and \(375 \mathrm{~mm} \mathrm{Hg}\) is cooled until the pressure is \(225 \mathrm{~mm} \mathrm{Hg}\). Assume that the volume remains constant.

Short answer

The final temperature is approximately \(-109.3^{\circ} \text{C}.\)

Step-by-step solution

Identify the Known Variables

The initial temperature is given as \(0^{\circ} \text{C}\), which can be converted to Kelvin by adding 273.15, giving \(273.15 \text{ K}\). The initial pressure \(P_1\) is \(375 \text{ mm Hg}\), and the final pressure \(P_2\) is \(225 \text{ mm Hg}\). The volume is constant at \(1.95 \text{ L}\).

Apply the Ideal Gas Law for Constant Volume

Since the volume is constant, we use the relation \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\), where \(T_1\) is the initial temperature and \(T_2\) is the final temperature in Kelvin. We can rearrange this to solve for \(T_2\): \(T_2 = \frac{P_2 \cdot T_1}{P_1}\).

Substitute the Values into the Equation

Substitute the known values into the equation: \(T_2 = \frac{225 \text{ mm Hg} \cdot 273.15 \text{ K}}{375 \text{ mm Hg}}\).

Calculate the Final Temperature in Kelvin

Perform the calculation: \(T_2 = \frac{225 \times 273.15}{375}\), which simplifies to \(T_2 \approx 163.893 \text{ K}\).

Convert the Final Temperature to Celsius

Convert the final temperature back to Celsius by subtracting 273.15 from the Kelvin temperature: \(T_{2, \text{C}} = 163.893 - 273.15 = -109.257 \text{ C}\).

Key concepts

Temperature Conversion

When working with gases and using the Ideal Gas Law, it's critical to use the absolute temperature scale, Kelvin, rather than Celsius. This is because the gas laws are directly proportional to Kelvin. For instance, if the temperature was given as 0°C, it needs to be converted to Kelvin by adding 273.15, thus making it 273.15 K.
This conversion ensures a positive, non-zero temperature, which is necessary because the Kelvin scale starts at absolute zero. Calculations with Celsius often require conversion to Kelvin to apply the Ideal Gas Law correctly. When you've completed calculations in Kelvin, you might need to convert back to Celsius for more intuitive temperature representation. To convert Kelvin back to Celsius, simply subtract 273.15 from your Kelvin value.

• Initial Temperature: Add 273.15 to Celsius to convert to Kelvin.
• Final Temperature: Calculate in Kelvin first, then subtract 273.15 to return to Celsius if needed.

Pressure Calculation

Understanding pressure calculations is essential when dealing with gases. The Ideal Gas Law can often involve different pressure units, but in most calculations involving gases, pressure is measured in units like atmospheres (atm), mmHg (millimeters of mercury), or Pascals (Pa).
In the given problem, the pressure is initially 375 mmHg and changes to 225 mmHg. We use these pressures because of the constant volume condition, as expressed in the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). This equation shows how at a constant volume, the ratio of pressure to temperature remains constant.

• Initial Pressure, \( P_1 \): The starting pressure (375 mmHg).
• Final Pressure, \( P_2 \): The pressure after cooling (225 mmHg).
• Maintain unit consistency: In calculations, ensure all pressures are in the same units for accurate results.

Constant Volume

In many gas-related problems, the assumption of a constant volume simplifies calculations. When a gas is kept at a constant volume, any change in pressure is directly related to its change in temperature as per the Ideal Gas Law. In our problem, the volume stays at 1.95 L throughout.
This stability allows the use of the pressure-temperature relation. When volume does not change, the mathematical relationship becomes straightforward as there's no need to account for it in your calculations. This way, we simplify the problem by using the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), purely focusing on how pressure and temperature interact.
Keeping volume constant might happen in real scenarios like tightly sealed containers or rigid gas tanks where the volume cannot change due to inflexible walls.

• Constant Volume: 1.95 L, unchanged throughout the calculation.
• Focus on Pressure-Temperature Relation: With volume constant, it's easier to see how pressure impacts temperature.