Problem 24 - General Chemistry: Principles and Modern Applications

Question

A 10.0 g sample of a gas has a volume of \(5.25 \mathrm{L}\) at \(25^{\circ} \mathrm{C}\) and \(762 \mathrm{mm} \mathrm{Hg} .\) If \(2.5 \mathrm{g}\) of the same gas is added to this constant 5.25 L volume and the temperature raised to \(62^{\circ} \mathrm{C},\) what is the new gas pressure?

Step-by-Step Solution

Answer

After solving the equations, the new gas pressure will be the value of \(P_2\) as found in Step 3.

Step 1: Finding the Number of Moles for Initial Conditions

Use the ideal gas law to calculate the initial number of moles. First, convert the temperature into Kelvin by adding 273.15 to the degrees Celsius. The formula was \(P_1V_1 = n_1RT\) where \(P_1 = 762 \mathrm{mm} \mathrm{Hg}\), \(V_1 = 5.25 \mathrm{L}\), \(R = 62.364 \mathrm{L.mmHg/mol.K}\) and \(T_1 = 25^{\circ} \mathrm{C} = 298.15 \mathrm{K}\). Solving for \(n_1\), we get \(n_1 = P1*V1/(R*T1)\)

Step 2: Adding More Gas

2.5g of the gas are added to the sample. The molar mass of the gas is given by relation between mass and moles from step 1. Calculate the number of moles of the added gas (\(n_2 = m/M\)), where m is the mass of added gas and M is the molar mass from step 1. The total number of moles (\(n_{total}\)) is the sum of the initial and the added moles, \(n_{total} = n_1 + n_2\)

Step 3: Finding the New Gas Pressure

The new pressure (\(P_2\)) is found by solving the ideal gas law again with the new temperature, volume and number of moles. We convert the new temperature to Kelvin, \(T_2 = 62^{\circ} \mathrm{C} = 335.15 \mathrm{K}\). The volume (\(V_2\)) remains the same at \(5.25 \mathrm{L}\). Then solve for \(P_2\), \(P_2 = n_{total}*R*T_2/V_2\)

Key Concepts

Mole concept

The mole concept is a fundamental chemistry principle that allows chemists to count particles at the atomic scale. It links the mass of a substance to the number of particles it contains. - One mole of any substance contains Avogadro's number of particles, approximately \(6.022 \times 10^{23}\).- In our exercise, we first determine the number of moles by using the Ideal Gas Law, which involves the mass and molar mass of the gas. To find the number of moles of the gas initially, we need the molar mass, which is calculated indirectly through rearranging the Ideal Gas Law:\(n_1 = \frac{P_1 \times V_1}{R \times T_1}\). After calculating \(n_1\), we use the given mass for the second portion of the gas added, and the molar mass previously calculated, to find \(n_2\). This is done using:\(n_2 = \frac{m}{M}\). Adding these gives the total moles present in the system, \(n_{total} = n_1 + n_2\), which plays a crucial role in determining the new conditions of the gas.

Gas pressure calculation

Gas pressure is a vital concept in understanding how gases behave in different conditions. It is the force that the gas exerts on the walls of the container per unit area. In this exercise, to calculate gas pressure, we need to understand the relationship provided by the Ideal Gas Law.- The formula is \(PV = nRT\), which connects pressure (P), volume (V), the mole number (n), the gas constant (R), and temperature (T).- The initial pressure is given by \(P_1 = \frac{n_1 \times R \times T_1}{V_1}\). Using this, we determine the initial conditions.- After adding more gas, and adjusting the temperature, we re-calculate using new variables: \(P_2 = \frac{n_{total} \times R \times T_2}{V_2}\).This step ensures that we apply calculated mole values and revised temperature to determine the updated pressure. The constant volume doesn't affect the calculations directly for pressure alterations, as it remains unchanged.

Temperature conversion

Temperature plays a significant role in gas laws, as it directly influences the energy and movement of gas molecules. The crucial detail is converting temperatures to Kelvin, which is the absolute temperature scale, crucial for gas calculations. - The conversion from Celsius to Kelvin is done by adding 273.15 to the Celsius reading. - For example, the initial temperature of 25°C is converted to 298.15 K, necessary for our calculations related to initial conditions. - Similarly, when the new temperature of 62°C is considered, it converts to 335.15 K. This higher Kelvin temperature reflects the increase in thermal energy, impacting pressure calculations when using the Ideal Gas Law. Always remember: physics demands Kelvin for temperature!