Question

Carbon dioxide in a gas cylinder of unchangeable volume is at a pressure of \(25.0 \mathrm{~atm}\) at \(25^{\circ} \mathrm{C}\). When placed in the sunlight on a hot summer day, the temperature increases to \(40^{\circ} \mathrm{C}\). What is the new pressure of the \(\mathrm{CO}_{2}\) ?

Short answer

The new pressure of the \(\mathrm{CO}_{2}\) gas when the temperature increases to \(40^{\circ} \mathrm{C}\) is \(26.3 \mathrm{~atm}\).

Step-by-step solution

Write down the Gay-Lussac's Law equation

The equation for Gay-Lussac's Law is given as: \(P_1/T_1 = P_2/T_2\) Here, \(P_1\) is the initial pressure, \(T_1\) is the initial temperature, \(P_2\) is the final pressure and \(T_2\) is the final temperature.

Convert the temperatures to Kelvin

As we know, to work with gas laws, we need to have temperatures in Kelvin. Let's convert the given temperatures from Celsius to Kelvin: Initial temperature (\(T_1\)): \(25^{\circ} \mathrm{C} + 273.15 = 298.15 \mathrm{K}\) Final temperature (\(T_2\)): \(40^{\circ} \mathrm{C} + 273.15 = 313.15 \mathrm{K}\)

Substitute the given values into the equation

Now, we will substitute the given values into the Gay-Lussac's Law equation: \(\frac{25.0 \mathrm{~atm}}{298.15 \mathrm{K}} = \frac{P_2}{313.15 \mathrm{K}}\)

Solve for the final pressure (\(P_2\))

To find the new pressure of the \(\mathrm{CO}_{2}\) gas, rearrange and solve for \(P_2\): \(P_2 = \frac{25.0 \mathrm{~atm} \times 313.15 \mathrm{K}}{298.15 \mathrm{K}}\) Now, calculate the new pressure: \(P_2 = 26.3 \mathrm{~atm}\)

State the answer

The new pressure of the \(\mathrm{CO}_{2}\) gas when the temperature increases to \(40^{\circ} \mathrm{C}\) is \(26.3 \mathrm{~atm}\).

Key concepts

Gas Laws

Gas laws describe how gases behave under different conditions of pressure, volume, and temperature. They are fundamental in understanding how gases react to changes in their environment. The most common gas laws include Boyle's Law, Charles's Law, and Gay-Lussac's Law, among others.

• Boyle's Law: This law explains that the pressure of a gas is inversely proportional to its volume when the temperature is held constant. So, if you decrease the volume of a gas, its pressure increases, and vice versa.


• Charles's Law: This states that the volume of a gas is directly proportional to its temperature when the pressure is kept constant. As temperature increases, so does the volume.


• Gay-Lussac's Law: This specific law highlights the direct relationship between pressure and temperature, when volume is constant, which is the focus of our original exercise.

Each of these laws provides insight into how gases behave, allowing for predictions about their reactions to changes in conditions.

Pressure-Temperature Relationship

The pressure-temperature relationship for gases is epitomized by Gay-Lussac's Law. According to this law, when a gas is held at a constant volume, the pressure will increase if the temperature increases. Conversely, the pressure will decrease if the temperature decreases.

In the context of the exercise, as the temperature of the \(\mathrm{CO}_2\\) increases from 25°C to 40°C, the pressure within the gas cylinder increases. This occurs because temperature is essentially a measure of kinetic energy: as temperature goes up, gas molecules move faster, colliding with the walls of the cylinder with more force and more frequently, thus increasing pressure.

This can be seen in the equation: \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]
where \(P_1\) and \(P_2\) are the initial and final pressures, while \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin. This direct relationship ensures predictable changes when temperatures vary.

Temperature Conversion

When dealing with gas laws, precise temperature measurement is crucial because the calculations require temperatures to be in Kelvin. The Kelvin scale is preferred in scientific calculations because it starts at absolute zero, the theoretical point where molecular motion ceases.

To convert from Celsius to Kelvin, you add 273.15 to the Celsius temperature. For example, converting 25°C results in:

• 25°C + 273.15 = 298.15 K

Similarly, converting 40°C:

• 40°C + 273.15 = 313.15 K

This conversion allows for accurate application in equations like the one used in Gay-Lussac's Law. Using Kelvin ensures that our measurements reflect true proportional changes in temperature, essential for calculations that involve gas behavior.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of understanding gas behavior. Although not directly used in Gay-Lussac's Law, understanding it provides deeper insight into how gases act. The law is usually presented as:\[ PV = nRT \]
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

• This equation combines the other gas laws, showing how pressure, volume, temperature, and number of particles are interconnected.

Even though the equation suggests an "ideal" gas — one that perfectly follows the rules thus described — it gives a good approximation for the behavior of real gases under many conditions. The simplicity of these relationships helps us predict how a gas will respond to changes in environment, such as pressure or temperature increases, like our earlier CO2 cylinder example.