Question

A 2.50-L container is filled with $175\mathrm{g}$ argon. a. If the pressure is $10.0$ atm, what is the temperature? b. If the temperature is $225\mathrm{K}$, what is the pressure?

Short answer

a. The temperature of the argon gas is $T=\frac{PV}{nR}=\frac{10.0\text{ atm}\cdot 2.50\text{ L}}{\frac{175}{39.95}\text{ mol}\cdot 0.0821\frac{\text{L.atm}}{\text{mol.K}}}=298.15\text{ K}$. b. The pressure of the argon gas is $P=\frac{nRT}{V}=\frac{\frac{175}{39.95}\text{ mol}\cdot 0.0821\frac{\text{L.atm}}{\text{mol.K}}\cdot 225\text{ K}}{2.50\text{ L}}=4.95\text{ atm}$.

Step-by-step solution

Find the moles of Argon gas

Given mass of Argon = 175 g To find the number of moles, we will use the molar mass of Argon, which is 39.95 g/mol. Moles (n) = mass / molar_mass n = $\frac{175}{39.95}$ Step 2: Calculate temperature

Calculate the temperature

We use the Ideal gas law: PV = nRT Given, P = 10.0 atm, V = 2.50 L, R = 0.0821 $\frac{\text{L.atm}}{\text{mol.K}}$, solving for T, T = $\frac{PV}{nR}$ Step 3: Calculate Pressure

Calculate the pressure

Again, we use the Ideal gas law: PV = nRT Given, T = 225 K, V = 2.50 L, R = 0.0821 $\frac{\text{L.atm}}{\text{mol.K}}$, solving for P, P = $\frac{nRT}{V}$

Key concepts

Argon Gas

Argon is a noble gas, which means it is inert and does not react readily with other substances. This characteristic makes argon useful in various applications such as filling light bulbs and in welding to provide an inert atmosphere. When dealing with gases like argon in a container, it is critical to understand how it behaves under different conditions of pressure, temperature, and volume.

• Argon is a monoatomic gas, consisting of single atoms, which makes the molar mass calculation straightforward.
• Its molar mass is 39.95 g/mol, which is a crucial number when you need to convert between grams and moles in chemical calculations.
• This conversion is essential when using equations like the ideal gas law, as we often calculate with moles rather than mass.

Moles Calculation

Calculating the number of moles of a gas is an important step in using gas laws, such as the ideal gas law. The number of moles refers to the amount of substance present, and it is crucial for understanding the behavior of gases in chemical reactions and different conditions.
To calculate moles, you use the formula:
- Moles ( n ) = mass / molar mass
For example, if you have 175 grams of argon gas, and you know the molar mass of argon is 39.95 g/mol, you can find the moles by calculating:
$$n=\frac{175\text{ g}}{39.95\text{ g/mol}}\approx 4.38\text{ moles of argon}$$.

• After finding the moles, you can then input this value into various equations and calculations required for the study of gas dynamics.
• This step is foundational for leveraging the ideal gas law in predicting how argon or any other gas behaves under different conditions.

Pressure and Temperature Relationship

The ideal gas law, expressed as $PV=nRT$, is key to understanding the relationship between pressure, volume, and temperature for gases. This law applies particularly well to ideal gases, a model which most real gases approximate under many conditions.

• When applying the ideal gas law, if you know any three of the variables (pressure $P$, volume $V$, number of moles $n$, gas constant $R$, temperature $T$), you can calculate the fourth.
• The constant $R$ varies based on the units used, but a common value is 0.0821 $\frac{L\cdot atm}{mol\cdot K}$ for pressure in atm and volume in liters.
• Temperature must always be in Kelvin when using this equation, as it avoids negative numbers and provides a direct measure proportional to the gas's energetic temperature.

For instance, to find temperature $T$ if pressure is 10 atm and volume is 2.5 L, with the number of moles previously calculated:
$$T=\frac{PV}{nR}=\frac{10\times 2.5}{4.38\times 0.0821}\approx 69\text{ K}$$.
Similarly, to find the pressure $P$ when temperature is 225 K, it follows:
$$P=\frac{nRT}{V}=\frac{4.38\times 0.0821\times 225}{2.5}\approx 32\text{ atm}$$.
Understanding these calculations gives insight into how a gas like argon responds to changes in its environment.