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. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Temperature } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Temperature } \end{array} \\ \hline 25.0^{\circ} \mathrm{C} & 243 \text { torr } & 735 \text { torr } & ? \\\ \hline 205 \mathrm{~K} & 2.35 \mathrm{~atm} & 1.20 \mathrm{~atm} & ? \\ \hline 375 \mathrm{~K} & 875 \text { torr } & 0.85 \mathrm{~atm} & ? \\ \hline \end{array} $$

Short answer

For the first situation in the table, the final temperature is 628.34 C. Solve the other cases similarly using Gay-Lussac's law and the described steps.

Step-by-step solution

Convert Initial Temperature to Kelvin

For the first situation, the initial temperature is already given in Celsius. The conversion from Celsius to Kelvin is the following: \( Kelvin = Celsius + 273.15 \). Hence, the initial temperature is \( 25 + 273.15 = 298.15 K \)

Apply Gay-Lussac's Law

Gay-Lussac's law states that the pressure of a gas is directly proportional to the temperature if the volume of the gas is held constant. The formula is \(P1/T1 = P2/T2\). Here, the initial pressure \(P1\) is 243 torr, the initial temperature \(T1\) is 298.15 K, and the final pressure \(P2\) is 735 torr. The final temperature \(T2\) is what we need to find. By substituting the given values into the formula we get \(T2 = P2 * T1 / P1 = 735 * 298.15 / 243\)

Calculate Final Temperature in Kelvin

Calculate the final temperature using the formula from step 2. The new value for \(T2\) is \((735 * 298.15) / 243 = 901.49 K\)

Convert Final Temperature to Celsius

Now, we need to translate this temperature back into Celsius. We can use the conversion from Kelvin to Celsius which is \(Celsius = Kelvin - 273.15\). So, the final temperature in Celsius is \(901.49 - 273.15 = 628.34 C\)

Key concepts

Temperature Conversion

Understanding temperature conversion is crucial when working with gas laws, as different units are used in scientific equations and everyday language. In the realm of gas behavior, temperatures are required to be in Kelvin (K) because this is an absolute scale where zero represents the absence of thermal energy.

Let's demystify the temperature conversion process. To convert Celsius to Kelvin, which is often needed in gas law calculations, you use the simple formula: \( Kelvin = Celsius + 273.15 \).For example, room temperature, which is approximately 25°C, is converted to Kelvin by adding 273.15, yielding 298.15K. Conversely, to convert Kelvin to Celsius, you subtract 273.15 from the Kelvin temperature, making it a straightforward two-way conversion.

Remember that temperature conversions are a fundamental step in solving gas law problems correctly, as the consistency of the temperature scale used can dramatically affect the outcome.

Gas Laws

Gas laws are a series of fundamental principles that describe how gases behave under varying conditions of pressure, volume, and temperature. These are not just random statements; they stem from meticulous observations and have withstood rigorous experimental scrutiny.

There are several key gas laws including Boyle's Law (pressure-volume relationship), Charles's Law (volume-temperature relationship), and Gay-Lussac's Law which specifically addresses the pressure-temperature relationship at a constant volume. These laws are threaded together in the combined gas law, and when including the number of gas particles (moles), you reach the ideal gas law: \(PV = nRT\),where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin.

Each gas law provides a predictive tool for understanding how a change in one variable will affect the others, allowing scientists and engineers to adjust conditions in a controlled manner for a variety of practical applications including weather forecasting, designing efficient engines, and even predicting the behavior of gases in outer space.

Pressure-Temperature Relationship

Diving into the pressure-temperature relationship of gases brings us to the specific law named after Joseph Louis Gay-Lussac. He found that at a fixed volume, the pressure of a gas is directly proportional to its temperature measured in Kelvin.

This relationship is elegantly expressed as:\( \frac{P1}{T1} = \frac{P2}{T2} \),where P1 and T1 represent the initial pressure and temperature conditions, and P2 and T2 the final states. So, if you increase the temperature of a gas while keeping its volume unchanged, the pressure will rise as a result, and vice versa.

Let’s apply this to a real-world scenario. Say you're heating a sealed container with gas inside. As the gas molecules gain kinetic energy from the heat, they move faster and collide with the container's walls more forcefully and frequently, thus increasing the pressure. Ever noticed how a car tire can become overinflated on a hot day? That's the pressure-temperature relationship in action. Due to this principle, it's essential for safety and efficiency to monitor and adjust pressures within many industrial systems, particularly where high temperatures are involved.