Question

Suppose that a 1.25-g sample of neon gas is confined in a 10.1-L. container at \(25^{\circ} \mathrm{C}\). What will be the pressure in the container? Suppose the temperature is then raised to \(50^{\circ} \mathrm{C}\). What will the new pressure be after the temperature is increased?

Short answer

The initial pressure in the container is calculated using the Ideal Gas Law and given parameters, which results in: P = \( \frac{(1.25/20.18) \ \mathrm{mol} \times 0.0821 \ \mathrm{L \cdot atm/mol \cdot K} \times 298.15 \ \mathrm{K}}{10.1 \ \mathrm{L}} \) After the temperature increase to 50°C (323.15 K), the new pressure is calculated using the relation: P_2 = P × \( \frac{323.15 \ \mathrm{K}}{298.15 \ \mathrm{K}} \)

Step-by-step solution

Convert mass of neon gas to moles

We have 1.25 grams of neon gas, and the molar mass of neon is 20.18 g/mol. To convert the mass of neon gas to moles, we use the following formula: moles of neon = mass of neon / molar mass of neon moles of neon = \( \frac{1.25 \ \mathrm{g}}{20.18 \ \mathrm{g/mol}} \)

Convert temperature in Celsius to Kelvin

Given temperature = 25°C, we have to convert the temperature to Kelvin using the following formula: Temperature in Kelvin = Temperature in Celsius + 273.15 T = 25 + 273.15 = 298.15 K For the increased temperature: T_new = 50 + 273.15 = 323.15 K

Calculate the pressure using Ideal Gas Law

Now, we have all the values to calculate the pressure P using the Ideal Gas Law equation (PV = nRT). For this exercise, we will use the Ideal Gas Constant R = 0.0821 L·atm/mol·K. P = \( \frac{nRT}{V} \) P = \( \frac{(1.25/20.18) \ \mathrm{mol} \times 0.0821 \ \mathrm{L \cdot atm/mol \cdot K} \times 298.15 \ \mathrm{K}}{10.1 \ \mathrm{L}} \)

Calculate the new pressure when temperature increases

As we know the volume doesn't change, and the moles of gas remain the same, we can calculate the new pressure using the following relation: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) \( P_2 = P_1 \times \frac{T_2}{T_1} \) P_2 = P × \( \frac{323.15 \ \mathrm{K}}{298.15 \ \mathrm{K}} \)

Key concepts

Pressure Calculation

Calculating the pressure of a gas involves using the Ideal Gas Law, which is described by the formula \( PV = nRT \). Here, \( P \) stands for pressure, \( V \) is the volume, \( n \) represents the moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

To calculate the initial pressure in our neon gas scenario, we first determine the number of moles from the given mass. Then, we substitute values for \( n \), \( R \), \( V \), and \( T \) into the formula to find \( P \). Remember that the ideal gas constant \( R \) is typically \( 0.0821 \ \mathrm{L \, atm/mol \, K} \) when using atmospheres and liters.

• We found the number of moles of neon, \( n = \frac{1.25}{20.18} \) mol.
• The initial volume \( V \) is given at \( 10.1 \) L.
• The temperature \( T \) must be in Kelvin.

By substituting these values into the equation \( P = \frac{nRT}{V} \), we can calculate the pressure of the gas in the container initially.

Temperature Conversion

In gas law calculations, temperature must always be expressed in Kelvin, which is the absolute temperature scale. This ensures the accuracy and consistency of the results since Kelvin does not allow for negative numbers, unlike Celsius which can have negative temperatures that make physical sense.

To convert a temperature in Celsius to Kelvin, you simply add 273.15.

• For example, to convert \( 25^{\circ} \mathrm{C} \) to Kelvin, use \( 25 + 273.15 = 298.15 \ \mathrm{K} \).
• Similarly, a temperature increase to \( 50^{\circ} \mathrm{C} \) translates to \( 323.15 \ \mathrm{K} \).

Temperature differences are preserved in Kelvin similarly to Celsius.

Understanding this conversion is crucial, as the gas laws are derived based on thermodynamic principles that work with absolute temperatures.

Moles of Gas

The concept of moles is fundamental in chemistry and is especially important in gas calculations. A mole is a unit of measurement for the amount of substance, based on the number of atoms in 12 grams of carbon-12. This number is Avogadro's number, which is approximately \( 6.022 \times 10^{23} \) particles per mole.

Using the formula \( n = \frac{\text{mass of gas}}{\text{molar mass}} \), we can convert the mass of a gas sample to moles, which is often a necessary step in gas law calculations. For neon, with a molar mass of \( 20.18 \ \mathrm{g/mol} \), we find the moles by dividing the mass of the neon sample by its molar mass.

• In this exercise, \( n = \frac{1.25 \ \mathrm{g}}{20.18 \ \mathrm{g/mol}} \), gives us the quantity in moles to use in subsequent calculations.

This conversion is crucial for determining other properties like pressure and volume in gas-related problems.