Question

Assume that the volume of a fixed amount of gas in a rigid container does not change. Calculate the temperature in degrees Celsius to which the gas would have to be changed to achieve the change in pressure shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Temperature } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Temperature } \end{array} \\ \hline 30.0^{\circ} \mathrm{C} & 1525 \text { torr } & 915 \text { torr } & ? \\\ \hline 250.0^{\circ} \mathrm{C} & 0.70 \mathrm{~atm} & 1042 \text { torr } & ? \\\ \hline 355 \mathrm{~K} & 500.0 \text { torr } & 1000.0 \text { torr } & ? \\ \hline \end{array} $$

Short answer

The final temperatures corresponding to the changes in pressure stated in the table for each row of data are as follows: -92.03°C, 749.45°C and 436.85°C.

Step-by-step solution

Calculate the final temperature for the first row of data

Use the formula of the law of Gay-Lussac, P1/T1 = P2/T2, rearranged to solve for T2, which is T2 = P2 * T1 / P1. Substitute the given values into the equation: T1 is the initial temperature (30.0°C), but you must convert it to Kelvin by adding 273.15, resulting 303.15 K. The initial pressure P1 is 1525 torr, and the final pressure P2 is 915 torr. And so, T2 = 915 * 303.15 / 1525, resulting in T2 = 181.12 K which in Celcius is -92.03°C.

Calculate the final temperature for the second row of data

Apply the same formula and conversion principles. Convert initial temperature from Celsius to Kelvin, resulting in 523.15 K. Also, it's very important to keep consistent units of pressure, therefore convert the initial pressure P1 from atm to torr: 1 atm = 760 torr, hence 0.70 * 760 = 532 torr. T2 = 1042 * 523.15 / 532, resulting in T2 = 1022.60 K, which is 749.45°C in Celsius.

Calculate the final temperature for the third row of data

Apply the same formula but no need to convert the initial temperature, as it is already in the proper unit (Kelvin). T2 = 1000 * 355 / 500, resulting in T2 = 710 K, which is 436.85°C in Celsius.

Key concepts

Gas Laws

Gas laws are a set of rules that describe the behavior of gases under various conditions. These laws help us understand how variables such as temperature, pressure, and volume interact in a gaseous system. Three of the most well-known gas laws are Boyle's Law, Charles's Law, and Gay-Lussac's Law. Here’s a quick overview:

• Boyle's Law: This law states that the pressure of a gas is inversely proportional to its volume when the temperature is constant. In other words, if the volume decreases, the pressure increases, and vice versa.


• Charles's Law: According to this law, the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is constant. So, if the temperature increases, so does the volume.


• Gay-Lussac's Law: This law maintains that the pressure of a gas is directly proportional to its temperature (in Kelvin) if the volume is constant, making it particularly useful for problems involving closed systems like rigid containers.

Understanding these laws is crucial as they form the foundation of thermodynamics. They are essential for calculating changes in state parameters, evident in many practical applications such as diving, aerodynamics, and even cooking!

Gay-Lussac's Law

Gay-Lussac's Law is specifically focused on the relationship between the pressure and temperature of a gas while keeping volume constant. The formula is expressed as \( \frac{P1}{T1} = \frac{P2}{T2} \). Importantly, temperatures must be measured in Kelvin for accuracy.

Here's how to use it:

• Identify initial and final pressures: Know your initial pressure \(P_1\) and final pressure \(P_2\).
• Convert temperatures to Kelvin: Initial temperature \(T_1\) needs to be converted into Kelvin by adding 273.15 to the Celsius temperature.
• Rearrange the formula: Solve for the unknown temperature \(T_2\): \(T_2 = \frac{P_2 \times T_1}{P_1}\).

By following these steps, we can predict how changes in temperature will affect the pressure of a gas within a closed system. This is particularly useful in situations where the gas can't expand or contract, such as in a sealed container.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is crucial for correctly applying gas laws, including Gay-Lussac's Law. The Kelvin scale is the standard unit of temperature for scientific calculations because it starts at absolute zero, the coldest possible temperature where particles theoretically stop moving.

To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature:
\[ K = °C + 273.15 \]
Conversely, to convert from Kelvin to Celsius, subtract 273.15 from the Kelvin temperature:
\[ °C = K - 273.15 \]
This conversion is straightforward yet essential, as incorrect temperature units will lead to inaccurate results in gas laws calculations. Always double-check your temperature values and ensure they're in Kelvin before performing calculations!

Pressure Units

Pressure units can often be a source of confusion due to the various scales available, such as torr, atmospheres (atm), and pascals (Pa). Understanding how to convert between these units is important when utilizing Gay-Lussac's Law or any gas law.

Here are some key conversions:

• From atm to torr: 1 atm equals 760 torr.
• From torr to atm: Divide the pressure in torr by 760.
• From atm to pascals: 1 atm equals 101,325 Pa.

Always ensure that pressures are consistently in the same units when performing calculations. In the exercise above, for instance, it's crucial to convert pressures appropriately in each row's data before using Gay-Lussac's Law. This keeps calculations accurate and ensures a coherent understanding of the gas behavior in the system.