Applying the Ideal Gas Law: Calculating Pressure in Atmospheres

The ideal gas law is a cornerstone principle in chemistry and physics, providing a way to understand the behavior of gases under different conditions. The law is widely used in scientific and engineering applications, from weather prediction to gas design in everyday devices.

Understanding the Ideal Gas Law

The ideal gas law is expressed by the equation:

PV nRT

Where:

V is the volume of the gas (in cubic meters), n is the number of moles of the gas, R is the ideal gas constant, which varies based on the units used, T is the temperature of the gas (in Kelvin).

Amontons' transformation, a variation of the ideal gas law, can solve for different variables:

P nRT/V

Calculating Pressure in Atmospheres

Let's consider a scenario where a 0.50 mol sample of nitrogen gas is contained in a 10.0 L container at a temperature of 298 K. We want to calculate the pressure in atmospheres exerted by the gas.

Given:

V 10 L (converted to 0.01 m3) n 0.50 mol R 8.314 J/(mol·K) or, for convenience, 8.314 m3Pa/(mol·K) T 298 K

Substituting these values into the equation:

P frac{nRT}{V} frac{0.50 text{ mol} times 8.314 text{ m}^3 text{Pa/(mol K)} times 298 text{ K}}{0.01 text{ m}^3} 123879 text{ Pa}

Given that 1 atmosphere (atm) is approximately 101325 Pa, the pressure can be converted to atmospheres by dividing the result by 101325:

P frac{123879 text{ Pa}}{101325 text{ Pa/atm}} 1.22 text{ atm}

Solving for Volume Using the Ideal Gas Law

Using the same ideal gas law equation, we can also solve for volume if we have the pressure, number of moles, and temperature. The equation rearranges to:

V frac{nRT}{P}

If we consider the second scenario where we find the volume using the same pressure of 1.22 atm, the value of R for atmospheres and liters is 0.08206 L·atm/(mol·K), the number of moles is 0.50 mol, and the temperature is 298 K:

V frac{0.50 text{ mol} times 0.08206 text{ L atm/(mol K)} times 298 text{ K}}{1.22 text{ atm}} 10.0 text{ L}

Understanding Temperature Conversion

The temperature in Kelvin is given as 298 K, which corresponds to 25°C. This conversion is important when working with the ideal gas law, as temperature in Kelvin is required for the equation to hold true. The Kelvin scale is absolute, and it is commonly used in scientific calculations to ensure that temperature differences are accurately represented.

Converting between Celsius and Kelvin:

T(K) T(°C) 273.15

Thus, 298 K 25°C 273.15.

The pressure in atmospheres is a convenient unit in many practical applications because it directly correlates with standard atmospheric pressure at sea level. Understanding and being able to calculate pressure in atmospheres is crucial for a wide range of scientific and engineering tasks, from designing pressure vessels to modeling atmospheric conditions.