Question

At \(27^{\circ} \mathrm{C}, 10.0\) moles of a gas in a \(1.50-\mathrm{L}\) container exert a pressure of 130 atm. Is this an ideal gas?

Short answer

To determine whether the gas is ideal or not, evaluate R from the ideal gas law equation with the given values. If the value matches the universal constant, then it's an ideal gas.

Step-by-step solution

Convert temperature to Kelvin

Convert the given \(27^{\circ}C\) to Kelvin by adding 273.15 \[ T = 27 + 273.15 = 300.15 K\]

Substitute given values into ideal gas law

Substitute the given values into the ideal gas law, \(PV = nRT\) and solve for R \[R = \frac{PV}{nT} = \frac{(130 \, \text{atm})(1.5 \, \text{L})}{(10.0 \, \text{mol})(300.15 \, \text{K})}\]

Evaluate R and determine if the gas is ideal

After evaluating the expression, if the calculated R-value is close to the known universal gas constant \((0.0821 \, \frac{L \cdot atm}{mol \cdot K})\), then the gas is ideal, otherwise it's not.

Key concepts

Universal Gas Constant

The universal gas constant, often symbolized by \( R \), is a crucial component of the ideal gas law. This constant links the molar values of pressure, volume, temperature, and amount of gas in a mathematical relationship. The ideal gas law is expressed as:

• \( PV = nRT \)

Here, each symbol plays an essential role:

• \( P \) represents the pressure of the gas
• \( V \) is the volume
• \( n \) denotes the number of moles of the gas
• \( T \) stands for the temperature in Kelvin

The gas constant, \( R \), generally takes the value \( 0.0821 \, \frac{L \, atm}{mol \, K} \) when pressure is measured in atmospheres (atm) and volume in liters (L). By evaluating the given variables against this constant, we can determine if a gas behaves ideally. If the calculated \( R \) matches the universal gas constant closely, the gas can be considered ideal.

Gas Pressure Calculation

Gas pressure is an important measure in determining the behavior of gases under various conditions. In the ideal gas law, pressure is denoted by \( P \) and is vital for calculating other parameters like the universal gas constant.
Pressure can be easily calculated using the ideal gas law formula. If you know the volume, number of moles, temperature, and the constant \( R \), you can rearrange the equation \( PV = nRT \) to solve for pressure \( P \):

• \( P = \frac{nRT}{V} \)

In our example, the pressure of the gas is already given as 130 atm. This can also be calculated from known conditions using the above formula, providing insight into whether the gas behaves ideally or not compared to theoretical predictions. Calculating pressure allows us to explore the gas's behavior under specific conditions and make predictions based on the ideal gas law.

Temperature Conversion to Kelvin

Converting temperature to Kelvin is a crucial step when working with the ideal gas law because the law requires temperature to be measured in an absolute scale. The Kelvin scale begins at absolute zero, which is the theoretical point where molecular movement stops.
To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For example, a temperature of \(27^{\circ} C\) converts to Kelvin as follows:

• \( T = 27 + 273.15 = 300.15 \, K \)

This conversion ensures that the temperature is in the correct unit for use in the ideal gas equation. Kelvin is the SI unit for temperature, making it universally applicable in scientific formulas such as the ideal gas law. Without converting to Kelvin, calculations within the ideal gas law would yield incorrect results, as the absolute scale is essential for accurate representation of thermal energy.