Problem 70 - Chemistry: An Atoms-Focused Approach

Question

At what temperature will 1.00 mol of an ideal gas in a \(1.00 \mathrm{L}\) container exert a pressure of 1.00 atm?

Step-by-Step Solution

Answer

Answer: Approximately 12.2 K.

Step 1: Identify the given values

In this problem, we know that: - P = 1.00 atm; - V = 1.00 L; - n = 1.00 moles; - R = 0.08206 L atm K⁻¹ mol⁻¹.

Step 2: Rearrange the Ideal Gas Law formula to solve for T

We want to find the temperature, T. To do that, we need to rearrange the Ideal Gas Law formula to isolate T: PV / (nR) = T.

Step 3: Substitute the given values into the formula and solve for T

Now, we can plug the given values into the rearranged formula: T = (1.00 atm × 1.00 L) / (1.00 mol × 0.08206 L atm K⁻¹ mol⁻¹).

Step 4: Calculate T

Perform the calculation to find T: T ≈ (1.00) / (0.08206). T ≈ 12.187 K. So, 1.00 mol of an ideal gas in a 1.00 L container will exert a pressure of 1.00 atm at approximately 12.2 K.

Key Concepts

Temperature Calculation Using the Ideal Gas Law

The Ideal Gas Law is crucial when calculating the temperature of gases under specific conditions. The formula for this law is given as:\[ PV = nRT \]Where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles.
• \(R\) is the ideal gas constant \(0.08206 \text{ L atm K}^{-1} \text{ mol}^{-1}\).
• \(T\) is the temperature in Kelvin.

To calculate temperature, \(T\), we rearrange the formula:\[ T = \frac{PV}{nR} \]You can use this formula to find out the temperature when specific values for pressure, volume, and number of moles are known. This method is practical and direct. Simply substitute the given values into the equation and perform the arithmetic calculations. Make sure that your pressure is in atmospheres, the volume is in liters, and that the gas constant is consistent with these units to ensure accurate results.
In our original problem, substituting known values as follows:\[ T = \frac{(1.00 \text{ atm} \times 1.00 \text{ L})}{1.00 \text{ mol} \times 0.08206 \text{ L atm K}^{-1} \text{ mol}^{-1}} \approx 12.2 \text{ K} \]Thus, the temperature at which the gas exerts a pressure of 1.00 atm is approximately 12.2 K.

Understanding the Mole Concept

The mole concept is a fundamental idea in chemistry. It allows us to relate the mass of a substance to the number of atoms or molecules it contains. A mole refers to \(6.022 \times 10^{23}\) units of a substance, also known as Avogadro's number.
This concept is essential when dealing with gases, as it provides a bridge between microscopic and macroscopic scales. In our Ideal Gas Law context, we use moles to quantify the number of gas molecules in a given sample, denoted by \(n\) in the formula \( PV = nRT \). When calculating the temperature, knowing that there are exactly \(1.00\) moles present, for instance, directly impacts the resultant temperature since it appears in the denominator of our rearranged formula \( T = \frac{PV}{nR} \).
Understanding moles helps in comprehending how much substance is present in a balanced chemical context, and when related to gases, it informs how a certain number of moles under given pressure and volume conditions relate to temperature.

Pressure and Volume Relationship in Gases

Pressure and volume are two of the most direct influencing factors in gas behavior, and they have an inverse relationship according to Boyle’s Law, which states that for a given amount of gas at a constant temperature, pressure is inversely proportional to volume:\[ P \propto \frac{1}{V} \]However, when temperature is introduced, as it is in the Ideal Gas Law, their relationship becomes linear when combined with mole number and constant. Here it's crucial to note how they both play out in the equation \( PV = nRT \).
A stronger grasp of this relationship shows how increasing the volume can lower pressure if temperature and the number of moles remain constant. Conversely, decreasing the volume tends to increase pressure in confined gases, depicting their natural tendency to expand or compress.
When configuring the Ideal Gas Law for temperature, as we've done here to find \( T = \frac{PV}{nR} \), both pressure and volume are directly proportional to the resulting temperature. Hence, understanding this relationship is key, especially when working backwards to determine either temperature or changes due to modifications in pressure or volume.