Question

A gas cylinder is filled with argon at a pressure of 177 atm and \(25^{\circ} \mathrm{C}\). What is the gas pressure when the temperature of the cylinder and its contents are heated to \(195^{\circ} \mathrm{C}\) by exposure to fire?

Short answer

The gas pressure when heated to 195°C is approximately 278.2 atm.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law states that for a given amount of gas, the pressure (P), volume (V), and temperature (T) are related by the equation: \[ PV = nRT \]where \(n\) is the number of moles and \(R\) is the universal gas constant. For this problem, we need to use the relation between pressure and temperature while keeping volume and the quantity of gas constant.

Use the Combined Gas Law

Since the amount of gas and volume are constant, the Combined Gas Law can simplify to: \[ \frac{P1}{T1} = \frac{P2}{T2} \]Here, \(P1 = 177 \text{ atm}\), \(T1 = 25^{\, \circ} \text{C}\), and \(T2 = 195^{\, \circ} \text{C}\).

Convert Temperatures to Kelvin

Convert both temperatures from Celsius to Kelvin using the conversion formula: \(T (K) = T (^{\, \circ}\text{C}) + 273.15\). Thus, \(T1 = 25 + 273.15 = 298.15 \text{ K}\) and \(T2 = 195 + 273.15 = 468.15 \text{ K}\).

Solve for the Unknown Pressure

Substitute the known values into the Combined Gas Law and solve for \(P2\): \[ \frac{177}{298.15} = \frac{P2}{468.15} \]Cross-multiplying to solve for \(P2\) gives \[ P2 = 177 \times \frac{468.15}{298.15} \approx 278.2 \text{ atm} \]

Key concepts

Combined Gas Law

When dealing with gases, it is crucial to understand how pressure, volume, and temperature relate to each other. The Combined Gas Law combines three individual gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. This law is written as: \[ \frac{P1V1}{T1} = \frac{P2V2}{T2} \] where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature in Kelvin. The equation shows us how two sets of conditions are related when the amount of gas is constant. In our problem, the volume and the amount of gas are constant, which simplifies the law to: \[ \frac{P1}{T1} = \frac{P2}{T2} \] This relationship helps us understand how pressure changes when temperature changes under constant volume.

Pressure-Temperature Relationship

The relationship between pressure and temperature is described by Gay-Lussac's Law, which states that pressure is directly proportional to temperature when volume is constant. This means if you increase the temperature of a gas, its pressure increases as well, provided the volume does not change. In mathematical terms, we write this relationship as: \[ \frac{P1}{T1} = \frac{P2}{T2} \] For example, in our problem, the initial pressure (\(P1\)) in the gas cylinder is 177 atm and the initial temperature (\(T1\)) is 25°C. Upon heating to 195°C, the new pressure (\(P2\)) can be found using the equation above. Because temperature and pressure are directly proportional, the gas pressure increases as the temperature rises.

Temperature Conversion

When using gas laws, it's essential to convert temperatures from Celsius to Kelvin. The Kelvin scale is used in these calculations because it provides an absolute reference point (0 K is absolute zero, where particles theoretically stop moving). The conversion formula is simple: \[ T (K) = T (^\text{C}) + 273.15 \] In our specific problem, the initial temperature in Celsius is 25°C, which converts to: \[ 25 + 273.15 = 298.15 \text{ K} \] Similarly, the heated temperature of 195°C converts to: \[ 195 + 273.15 = 468.15 \text{ K} \] By converting temperatures to Kelvin, we ensure accurate and consistent use of the gas laws in solving for the new pressure in the cylinder.