Question

A gas sample containing \(1.50 \mathrm{~mol}\) at \(25^{\circ} \mathrm{C}\) exerts a pressure of 400 . torr. Some gas is added to the same container and the temperature is increased to \(50 .{ }^{\circ} \mathrm{C}\). If the pressure increases to 800 . torr, how many moles of gas were added to the container? Assume a constant-volume container.

Short answer

Approximately 0.603 moles of gas were added to the container.

Step-by-step solution

Write the Ideal Gas Law equation for the initial state

The initial state of the gas is given as: n=1.50 mol, T=25°C, and P=400 torr. We will convert temperature to Kelvin by adding 273.15 to Celsius temperature. Convert the given pressure to atm by dividing by 760, since 1 atm = 760 torr. Initial condition: \(P_1 = \frac{400}{760}atm\) \(T_1 = 25 + 273.15 = 298.15\) \(n_1 = 1.5\) From the Ideal Gas Law, we can write the equation for the initial state: \((P_1V) = n_1RT_1\)

Write the Ideal Gas Law equation for the final state

The final state of the gas is given as: T=50°C and P=800 torr. The number of moles in the final state is not given, so we will use \(n_2\) to represent the initial moles plus the added moles: \(n_2 = n_1 + x\), where x is the moles of gas added. Convert the final temperature to Kelvin and final pressure to atm: \(P_2 = \frac{800}{760}atm\) \(T_2 = 50 + 273.15 = 323.15\) From the Ideal Gas Law, we can write the equation for the final state: \((P_2V) = (n_1 + x)RT_2\)

Solve the equations

Since the volume of the container remains constant, we can divide the final state equation by the initial state equation: \[\frac{P_2V}{P_1V} = \frac{(n_1 + x)RT_2}{n_1RT_1}\] The volume V and gas constant R can be cancelled out, and we are left with: \[\frac{P_2}{P_1} = \frac{(n_1 + x)T_2}{n_1T_1}\] Now substitute the given values into the equation: \[\frac{\frac{800}{760}}{\frac{400}{760}} = \frac{(1.5 + x)(323.15)}{1.5(298.15)}\]

Find the moles of gas added

Simplify the equation and solve for x (moles of gas added): \[2 = \frac{(1.5 + x)(323.15)}{1.5(298.15)}\] Solve for x: \[(1.5 + x) = 2\times 1.5 \times \frac{298.15}{323.15}\] \[x = 1.5(\frac{596.3}{323.15} - 1)\] Now, calculate the value for x: \[x \approx 0.603\] So, approximately 0.603 moles of gas were added to the container.

Key concepts

Gas Pressure

Gas pressure, an essential concept for any student studying the behavior of gases, refers to the force that the gas exerts on the walls of its container. It’s measured in units such as atmospheres (atm), torr, or pascals (Pa). In the context of the Ideal Gas Law, which relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas, gas pressure serves as a key variable.

Understanding the relationship between gas pressure and the other variables is fundamental. For instance, in the given exercise, the pressure of a gas sample is initially at 400 torr and subsequently increases to 800 torr. This change in pressure is directly proportional to the number of moles and temperature, as long as the volume is constant, as stated by the Ideal Gas Law.

When solving problems involving pressure, it's important to ensure units are consistent, commonly converting torr to atm by dividing by the standard conversion factor (760 torr = 1 atm). This attention to detail facilitates the use of the Ideal Gas Law and leads to accurate calculations.

Moles of Gas

In the realm of chemistry, the concept of moles is akin to counting entities, just as one could count apples or oranges. Specifically, a mole of gas refers to Avogadro's number (approximately 6.022 x 10²³) of molecules. This counting allows chemists to relate amounts in the macroscopic world to the atomic scale.

The Ideal Gas Law makes it clear that the amount of substance, in moles, is pivotal for understanding gas behavior. In our problem, we start with 1.50 moles of gas and are asked to find out how many moles were added, denoted by 'x,' when the pressure and temperature change. This variable 'x' represents the unknown quantity that we're solving for, demonstrating how changes in the number of moles affect the overall state of the gas.

The step-by-step problem solving first identifies the initial moles (_1) and then introduces the unknown additional moles (_2 = n_1 + x). By manipulating the Ideal Gas Law equations for initial and final states, and maintaining other conditions like volume and temperature consistent or accounted for, you can solve for the quantity 'x.' It denotes a bridge between the theoretical understanding of moles and practical application in real-world scenarios.

Gas Temperature

Temperature is a crucial factor when dealing with gases, as it is a measure of the average kinetic energy of the gas molecules. In the context of the Ideal Gas Law (PVRT), temperature must always be expressed in Kelvin (K), which is an absolute temperature scale. This conversion is vital for calculations and is done by adding 273.15 to the Celsius temperature.

In the exercise, temperature initially is 25°C (T_1), which converts to 298.15 K, and then it rises to 50°C (T_2), or 323.15 K. The increase in temperature will cause the gas particles to move more vigorously, which, if the volume is kept constant, leads to an increase in pressure.

When calculating how various conditions affect gases, temperature is a vital aspect that can change outcomes significantly. A rise in temperature, assuming a constant volume, will require more space per gas particle, therefore, under such constraints, it leads to higher pressure. Remarkably, the Ideal Gas Law allows us to predict and quantify these changes mathematically, as observed when we use the changing temperatures in the problem at hand to determine the number of moles of gas added.