Question

A scuba diver's tank contains \(0.29 \mathrm{~kg}\) of \(\mathrm{O}_{2}\) compressed into a volume of \(2.3\) L. (a) Calculate the gas pressure inside the tank at \(9{ }^{\circ} \mathrm{C}\). (b) What volume would this oxygen occupy at \(26^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~atm}\) ?

Short answer

The gas pressure inside the tank is approximately \(88.79 atm\). The oxygen would occupy approximately \(218.5 L\) at \(26^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~atm}\).

Step-by-step solution

Convert temperature to Kelvin

To work with the Ideal Gas Law, we need to convert the given temperatures from Celsius to Kelvin. The conversion formula is: \(K = °C + 273.15\) For part (a): \(T_1 = 9°C + 273.15 = 282.15 K\) For part (b): \(T_2 = 26°C + 273.15 = 299.15 K\)

Calculate the number of moles of oxygen

We are given the mass of oxygen (\(0.29 kg\)) in the tank. To calculate the number of moles, we will use the molar mass of oxygen, which is approximately \(32 \frac{g}{mol}\). First, we need to convert the mass from kilograms to grams: \(0.29 kg \times 1000\frac{g}{kg} = 290 g\) Now, we can find the number of moles (\(n\)): \(n = \frac{mass}{molar~mass} = \frac{290 g}{32 \frac{g}{mol}} \approx 9.06 mol\)

Calculate the gas pressure inside the tank (part a)

Using the Ideal Gas Law (\(PV = nRT\)), we can now find the pressure inside the tank. We know the volume of the tank (\(2.3 L\)), the number of moles (\(9.06 mol\)), the temperature (\(282.15 K\)), and the ideal gas constant (\(R = 0.0821 \frac{L \cdot atm}{mol \cdot K}\)). We need to solve for the pressure (\(P\)): \(P = \frac{nRT}{V} = \frac{(9.06 mol)(0.0821 \frac{L \cdot atm}{mol \cdot K})(282.15 K)}{2.3 L} \approx 88.79 atm\) The gas pressure inside the tank is approximately \(88.79 atm\).

Calculate the volume of oxygen at different conditions (part b)

In this part, we need to find the volume that the oxygen would occupy at \(26^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~atm}\). For this, we'll use the combined gas law formula, given by: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) We know the initial pressure \(P_1 = 88.79 atm\), initial volume \(V_1 = 2.3 L\), initial temperature \(T_1 = 282.15 K\), final pressure \(P_2 = 0.95 atm\), and final temperature \(T_2 = 299.15 K\). We want to find the final volume \(V_2\). To find \(V_2\), we can rearrange the combined gas law formula: \(V_2 = V_1\frac{P_1}{P_2}\frac{T_2}{T_1} = (2.3 L)\frac{88.79 atm}{0.95 atm}\frac{299.15 K}{282.15 K} \approx 218.5 L\) The oxygen would occupy approximately \(218.5 L\) at \(26^{\circ} \mathrm{C}\) and \(0.95 \mathrm{~atm}\).

Key concepts

Combined Gas Law

Understanding the Combined Gas Law is crucial for dealing with scenarios where pressure, volume, and temperature of a gas are changing. This law is a combination of three fundamental gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law, and it enables the calculation of the state of a gas under varying conditions.
The formula for the Combined Gas Law is \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\). Here, \(P\), \(V\), and \(T\) represent the pressure, volume, and temperature of the gas, respectively, and the subscripts "1" and "2" denote the initial and final states of the gas.
This law helps us solve problems involving **changing conditions** of a gas, similar to how we calculate the final volume of the oxygen in the scuba tank when its temperature and pressure change. Remember, all temperatures must be converted to Kelvin when using this formula.

gas pressure calculation

Gas pressure calculation is fundamental in chemistry and physics, particularly when using the Ideal Gas Law. This law relates the pressure, volume, and temperature of a gas to the number of moles present. The formula is \(PV = nRT\), where:

• \(P\) is the pressure of the gas,
• \(V\) is its volume,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant (0.0821 L·atm/mol·K),
• and \(T\) is the temperature in Kelvin.

To calculate the gas pressure, we rearrange the formula to \(P = \frac{nRT}{V}\).
Substituting the known values such as temperature, volume, moles, and the gas constant, we can find the gas pressure inside a container, like in the scuba tank example. This fundamental calculation helps us understand how gases behave under different conditions.

temperature conversion

When working with gas laws, converting temperature from degrees Celsius to Kelvin is essential. This is because all gas law equations require the absolute temperature scale. The conversion formula is simple:
\(K = °C + 273.15\).
Kelvin is the SI unit for temperature, which begins at absolute zero, the theoretical point where particles have minimal thermal motion.
By converting Celsius to Kelvin, it becomes straightforward to insert temperatures into equations like the Ideal Gas Law or the Combined Gas Law without altering the relationship between pressure, volume, and temperature.
Always remember to check your units, because using Celsius in these equations will lead to incorrect results.

mole calculation

Accurately calculating moles is essential for understanding gas behaviors using the Ideal Gas Law. Moles measure the amount of substance, and in the context of gases, they help relate mass to volume and pressure.
To calculate moles from a given mass, use the formula:

• \(n = \frac{mass}{molar~mass}\).

The molar mass is the weight of one mole of a substance, often found on the periodic table. For example, oxygen has a molar mass of approximately 32 g/mol.
By using this formula, you can convert mass to moles, crucial for calculations involving gas pressure or volume changes. This conversion lays the groundwork for applying other gas laws accurately.
In our exercise, converting 0.29 kg of oxygen to moles helps determine how it will behave under different conditions in the tank.