Question

Aerosol cans carry clear warnings against incineration because of the high pressures that can develop if they are heated. Suppose that a can contains a residual amount of gas at a pressure of 755 mmHg and a temperature of \(25^{\circ} \mathrm{C} .\) What would the pressure be if the can were heated to \(1155^{\circ} \mathrm{C}\) ?

Short answer

The pressure inside the aerosol can, when heated to 1155°C, would be approximately 3612.28 mmHg.

Step-by-step solution

Convert the Initial Temperature to Kelvin

To use the gas laws, temperatures must be in Kelvin. Convert the initial temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. Initial temperature in Kelvin (T1) is: T1 = 25°C + 273.15 = 298.15 K.

Convert the Initial Pressure to Atm

The pressure needs to be in atmospheres (atm) for the calculations using the ideal gas law. Convert the initial pressure from mmHg to atm using the conversion factor 1 atm = 760 mmHg. Initial pressure in atm (P1) is: P1 = 755 mmHg * (1 atm / 760 mmHg) = 0.99342 atm.

Convert the Final Temperature to Kelvin

The final temperature must also be in Kelvin. Convert the final temperature from Celsius to Kelvin. Final temperature in Kelvin (T2) is: T2 = 1155°C + 273.15 = 1428.15 K.

Apply Gay-Lussac's Law

Gay-Lussac's Law defines the direct relationship between pressure and temperature of a gas at constant volume. It states that P1/T1 = P2/T2. Using this law, solve for the final pressure P2. P2 = (P1 * T2) / T1 Substitute the values of P1, T1, and T2 into the equation to calculate P2.

Calculate the Final Pressure

Using the equation from Step 4, calculate the final pressure P2. P2 = (0.99342 atm * 1428.15 K) / 298.15 K = 4.753 atm.

Convert the Final Pressure Back to mmHg

Since the initial pressure was given in mmHg, convert the final pressure back to mmHg using the conversion factor 1 atm = 760 mmHg. Final pressure in mmHg (P2) is: P2 = 4.753 atm * 760 mmHg/atm = 3612.28 mmHg.

Key concepts

Gas Pressure Calculations

Understanding gas pressure calculations is essential in chemistry, particularly when working with gas laws like Gay-Lussac's Law. Gas pressure is defined as the force exerted by gas molecules when they collide with the walls of their container. It is a critical factor in determining the behavior of gases under various conditions.

The problem from the textbook illustrates a real-world scenario where gas pressure increases due to heating, as seen with an aerosol can. To calculate gas pressure, standard units like atmospheres (atm), millimeters of mercury (mmHg), or pascals (Pa) are used depending on the context. In our case, the initial pressure is given in mmHg and later converted to atm for ease of calculation using Gay-Lussac's Law. Knowing how to convert between different pressure units is vital for accurate calculations.

When gas is heated at constant volume, its pressure typically increases, which can be dangerous in confined spaces such as an aerosol can. In this example, applying Gay-Lussac's Law allows us to find the new pressure after a temperature change. Remember, although the calculations might seem straightforward, error in unit conversion, for instance, from mmHg to atm or vice versa, could lead to incorrect results, an aspect often overlooked by students.

Temperature Conversion Kelvin

The temperature conversion to Kelvin is a fundamental step in working with gas laws and understanding the behavior of gases. Kelvin is the unit of temperature in the International System of Units (SI), and it's crucial for gas calculations because it is an absolute scale, meaning it starts at absolute zero, the theoretical point where particles have minimal thermal motion.

To convert Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature. This step is indispensable as most gas law formulas including the ideal gas law and Gay-Lussac's Law require temperature to be in Kelvin for the equations to work correctly. For instance, converting the initial temperature of the aerosol can from Celsius to Kelvin allows us to use Gay-Lussac's Law to calculate how much the pressure will increase when the can's temperature rises to a dangerous level if it were incinerated.

It's a common mistake to overlook this conversion, which is not just adding 273; the decimal .15 is crucial for precise calculations. Consequently, educating students about the importance of correct temperature conversion is part of ensuring their success in solving gas-related problems.

Ideal Gas Law Principles

The ideal gas law principles are foundational in understanding how gases behave under various conditions of pressure, volume, temperature, and amount. The ideal gas law is an equation of state for an ideal gas, which is a hypothetical gas that perfectly follows the kinetic-molecular theory. The law is usually expressed as PV = nRT, where P stands for pressure, V for volume, n for the number of moles, R for the universal gas constant, and T for temperature in Kelvin.

The ideal gas law synthesizes the combined gas law from Boyle's, Charles's, and Gay-Lussac's laws, providing a more comprehensive tool to predict and explain the behavior of gases. For instance, it helps in predicting the amount of gas required to achieve a certain pressure at a given temperature and volume, or how the pressure of a gas changes with the volume at constant temperature.Although real gases do not follow the ideal gas law perfectly, the principles offer significant insight, especially under normal conditions. It is critical for students to understand the ideal gas law principles to reason how real gases would deviate from ideal behavior at high pressures or low temperatures where intermolecular forces and the size of the gas molecules can no longer be ignored.