Problem 18 - Chemistry Matter and Change

Question

Calculate A rigid plastic container holds 1.00 L of methane gas at 660 torr pressure when the temperature is \(22.0^{\circ} \mathrm{C}\) . How much pressure will the gas exert if the temperature is raised to \(44.6^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The final pressure is 710.19 torr.

Step 1: Convert Temperatures to Kelvin

To use the ideal gas law, we need to convert the given temperatures from Celsius to Kelvin. The conversion formula is \( T(K) = T(^{\circ}C) + 273.15 \). For \(22.0^{\circ}C\), the temperature in Kelvin is \(22.0 + 273.15 = 295.15\,K\). For \(44.6^{\circ}C\), the temperature in Kelvin is \(44.6 + 273.15 = 317.75\,K\).

Step 2: Apply Gay-Lussac's Law

Since the volume of the container is constant, Gay-Lussac's Law, which involves pressure and temperature, is applicable. The law states that \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature.

Step 3: Substitute Known Values

Substitute the known values into Gay-Lussac's Law: \( \frac{660\, \text{torr}}{295.15\, \text{K}} = \frac{P_2}{317.75\, \text{K}} \).

Step 4: Solve for Final Pressure \(P_2\)

Cross-multiply to solve for \(P_2\): \( P_2 = \frac{660\, \text{torr} \times 317.75\, \text{K}}{295.15\, \text{K}} \). Calculating this gives \( P_2 = 710.19\, \text{torr} \).

Key Concepts

Gay-Lussac's Law

In the realm of gas laws, Gay-Lussac's Law is a crucial principle. It explains how pressure and temperature of a gas in a sealed container relate to each other.
This law holds true when the volume remains constant, like in a rigid plastic container with methane gas we considered in our example.
Gay-Lussac’s Law can be expressed mathematically as \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, \(P_1\) and \(T_1\) are the initial pressure and temperature, while \(P_2\) and \(T_2\) correspond to the pressure and temperature after a change.
This proportion shows us that if the temperature of gas increases, the pressure will also increase, provided the volume stays the same.

• This relationship is direct, meaning they increase and decrease together.
• Always remember that temperatures must be in Kelvin before applying the law.

This is how we can predict the new pressure of a gas when its temperature changes without altering the container’s volume.

Temperature Conversion

Temperature conversion is an essential step in dealing with gas laws. As we have seen, most gas equations require temperatures in Kelvin, not Celsius.
The Kelvin scale is an absolute temperature scale where absolute zero is 0 K, unlike Celsius.
This conversion is straightforward: simply add 273.15 to the Celsius temperature to get Kelvin. For instance:

• For \(22.0^{\circ}C\), converting gives \(22.0 + 273.15 = 295.15\, K\).
• For \(44.6^{\circ}C\), this becomes \(44.6 + 273.15 = 317.75\, K\).

Because Kelvin is directly related to energy, it is the appropriate unit for gas law equations.
Remember to always convert Celsius to Kelvin before plugging temperatures into any equations involving gas laws.

Pressure Calculation

Pressure calculation in gas laws helps us understand how gases behave under various conditions. When you know how temperature and other factors change, you can calculate new pressure values using formulas like Gay-Lussac’s Law.
In this example, after converting temperatures to Kelvin, you insert those values into the equation
\( \frac{660\, \text{torr}}{295.15\, \text{K}} = \frac{P_2}{317.75\, \text{K}} \).
By cross-multiplying, you solve for \(P_2\):
\[ P_2 = \frac{660\, \text{torr} \times 317.75\, \text{K}}{295.15\, \text{K}} \]
Computing this results in \(P_2 \approx 710.19\, \text{torr}\).
Thus, we see that an increase in temperature causes an increase in pressure, as expected in a closed system with constant volume.
This calculation shows how temperature and pressure are linked, helping you predict changes in gas behavior with variable conditions.