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

No, the gas is not ideal because the calculated \( PV \) differs from \( nRT \).

Step-by-step solution

Understand the Ideal Gas Law

The ideal gas law is written as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Convert Temperature to Kelvin

The temperature is given in Celsius. Convert it to Kelvin by adding 273.15: \[ T = 27^{\circ}C + 273.15 = 300.15 \text{ K} \]

Insert Values into the Ideal Gas Law

In the ideal gas law formula \( PV = nRT \), insert the given values: \[ P = 130 \text{ atm}, \quad V = 1.50 \text{ L}, \quad n = 10.0 \text{ moles}, \quad T = 300.15 \text{ K} \]Use \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \).

Calculate the Right Side of the Equation

Calculate \( nRT \):\[ nRT = 10.0 \text{ mol} \times 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \times 300.15 \text{ K} = 246.67665 \text{ L atm} \]

Calculate the Left Side of the Equation

Calculate \( PV \):\[ PV = 130 \text{ atm} \times 1.50 \text{ L} = 195 \text{ L atm} \]

Compare PV and nRT

Compare the calculated values of \( PV \) and \( nRT \):* \( PV = 195 \text{ L atm} \)* \( nRT = 246.67665 \text{ L atm} \)Since \( PV eq nRT \), the gas does not behave as an ideal gas under these conditions.

Key concepts

Pressure

Pressure is a force applied over a certain area. In terms of gases, it's the force that the gas molecules exert when they collide with the walls of their container.
When dealing with the ideal gas law, the pressure must be in a standardized unit, such as atmospheres (atm). This is because it needs to correspond with the other constants used in calculations, like the gas constant.
In our exercise, we see how a gas in a container with a specific volume (1.50 L) exerts a particular amount of pressure (130 atm). The force exerted by the gas particles is affected by various factors including temperature and the number of molecules present.

• Higher temperature can increase pressure because particles move faster and collide more.
• Adding more moles of gas increases pressure due to more particle collision with walls.
• Pressure and volume are inversely related if temperature and moles of gas are constant (Boyle’s Law).

Temperature conversion

Often, temperature needs to be converted from Celsius to Kelvin in chemistry problems that involve gas laws. Kelvin is the standard temperature scale used because it starts at absolute zero, the theoretically coldest point where all molecular motion stops, ensuring all values are positive.
The conversion is simple: you add 273.15 to the Celsius temperature. This ensures the accuracy needed in calculations involving gas laws like the Ideal Gas Law.
For our example, the temperature of 27°C was converted as follows:

• Formula: \[ T(K) = T(°C) + 273.15 \]
• Converted: \[ 27°C + 273.15 = 300.15 ext{ K} \]

Gas constant

The gas constant, denoted as \( R \), is a crucial component of the ideal gas law equation \( PV = nRT \). It serves as a bridge, allowing different units to fit together in a single equation.
For ideal gas calculations, \( R \) typically equals 0.0821 L atm K^{-1} mol^{-1}. This value ensures consistency among variables (pressure, volume, temperature in Kelvin, moles) in calculations.
Adjusting \( R \) would allow you to use different units for pressure or volume if needed, but it's essential to maintain consistency for accurate calculations. This is why it's vital to check unit compatibility when solving problems using the ideal gas law.

Moles

Moles are a basic unit in chemistry representing a quantity of substance. Specifically, one mole contains Avogadro's number of particles, which are approximately \( 6.022 imes 10^{23} \) particles of the substance.
In gas law calculations, moles indicate the amount of gas present. This is crucial because it directly influences the pressure exerted by the gas in a container. Knowing the number of moles helps predict how the gas will behave under different conditions based on the Ideal Gas Law.
In our scenario, we are given 10.0 moles of gas. Alongside pressure (130 atm), volume (1.50 L), and temperature (300.15 K), these moles help us determine whether the gas behaves ideally or not, by comparing actual pressure-volume work against theoretical calculations.