Question

Oxygen gas having a volume of \(1130 \mathrm{~cm}^{3}\) at \(42.0^{\circ} \mathrm{C}\) and a pressure of \(101 \mathrm{kPa}\) expands until its volume is \(1530 \mathrm{~cm}^{3}\) and its pressure is \(106 \mathrm{kPa}\). Find \((a)\) the number of moles of oxygen in the system and ( \(b\) ) its final temperature.

Short answer

The number of moles of oxygen in the system is calculated by substituting the appropriate values into the rearranged ideal gas equation. Similarly, the final temperature is calculated by substituting the values, including the value of n computed earlier, into a different form of the same equation. Both are found to the correct significant figures using a calculator.

Step-by-step solution

Convert given parameters to appropriate units and calculate initial temperature in Kelvin

Convert volume from \(cm^3\) to \(m^3\) by multiplying with \(1e-6\), so \(1130 cm^3\) becomes \(1.13e-3 m^3\). Convert pressure from kPa to Pa by multiplying with \(1e3\), so \(101 kPa\) becomes \(101e3 Pa\). Finally, convert the initial temperature from Celsius to Kelvin by adding 273.15 to \(42^{\circ} C\), which gives \(315.15 K\)

Calculate number of moles (n)

Rearranging the ideal gas equation \(PV = nRT\) to solve for n, we get \(n = PV/RT\). Substitute P = \(101e3 Pa\), V = \(1.13e-3 m^3\), R = \(8.314 J/(mol*K)\) and T = \(315.15 K\) to calculate n. This will give us the number of moles of oxygen gas in the system.

Convert final parameters to appropriate units

Convert the final volume and pressure to \(m^3\) and Pa respectively just as in Step 1. So, \(1530 cm^3\) becomes \(1.53e-3 m^3\) and \(106 kPa\) becomes \(106e3 Pa\)

Calculate final temperature (T)

Rearranging the ideal gas equation \(PV = nRT\) to solve for T, we get \(T = PV/nR\). Substitute P = \(106e3 Pa\), V = \(1.53e-3 m^3\), n as the number of moles calculated in Step 2, and R = \(8.314 J/(mol*K)\) to find the final temperature in Kelvin.

Key concepts

Understanding the Ideal Gas Equation

The ideal gas equation is a crucial tool in understanding the behavior of gases under various conditions. It is expressed as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

The ideal gas law allows us to calculate any one of these variables if the others are known. Since it relates pressure, volume, temperature, and moles, we often need to rearrange the equation depending on which variable we need to solve for. For instance, to find the number of moles, we rearrange the equation to \(n = \frac{PV}{RT}\). It is important to ensure all units are consistent; \(P\) should be in Pascals, \(V\) in cubic meters, and \(T\) in Kelvin. The gas constant \(R\) has a value of \(8.314 \frac{J}{mol\cdot K}\) when dealing with these units.

When applying the ideal gas law to real-world problems, it is necessary to begin by converting all measurements to the appropriate units to avoid any errors during calculations. The law assumes ideal conditions where gas molecules don't interact with each other, and in many cases, actual gases approximate this behavior well enough to use the ideal gas law for practical calculations.

Calculating the Moles of a Gas

In chemistry, the mole is a fundamental unit for quantifying the amount of substance. Calculating the number of moles in a gas using the ideal gas law involves rearranging the equation to solve for \(n\). By following this approach, we can directly relate the measurable properties of the gas, such as pressure \(P\), volume \(V\), and temperature \(T\), to the number of moles present.

For the oxygen gas in our problem, we use the rearranged equation \(n = \frac{PV}{RT}\). After converting all units to the standard SI units, we plug the values into the equation and find the number of moles. It's essential to use Kelvin for temperature and not degrees Celsius. Accuracy in this step is crucial, as any mistakes will affect subsequent calculations, such as determining the final temperature.

Understanding the mole concept allows us to bridge the gap between the macroscopic properties we observe (like pressure and volume) and the microscopic properties of individual gas particles. This connection makes mole calculations a fundamental aspect of chemical problem-solving.

Temperature Conversion in Gas Law Calculations

Temperature conversion is a basic but essential skill in dealing with gas laws, including the ideal gas equation. Since these equations require the temperature to be in Kelvin, converting from Celsius to Kelvin is a common prerequisite step.

To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is because 0 degrees Celsius is equivalent to 273.15 Kelvin—the two scales offset by this amount. The conversion ensures that we're working with an absolute temperature scale where zero represents the absolute zero of temperature, the point at which molecular motion theoretically stops.

For example, in the problem given, the initial temperature of the oxygen gas is \(42.0^\circ C\); adding 273.15 gives us the temperature in Kelvin, which is \(315.15 K\). This conversion is critical in gas law problems to ensure accurate calculations. Remember, temperature directly influences the pressure and volume of a gas, so using the correct temperature scale in calculations is essential for the correct determination of other gas properties.