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Understanding the concept of molar mass is crucial when dealing with gases and their reactions. Molar mass, often expressed in units of grams per mole (g/mol), is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. For diatomic oxygen, \( ext{O}_2\), the molar mass is calculated by multiplying the atomic mass of an oxygen atom (approximately 16 g/mol) by 2. Thus, the molar mass of \( ext{O}_2\) is 32 g/mol.
In this exercise, we have a mass of \(0.29 ext{ kg} = 290 ext{ g}\) of \( ext{O}_2\). To find out how many moles of oxygen are in the scuba diver's tank, we use the formula:

• Number of moles \(n = \frac{290 ext{ g}}{32 ext{ g/mol}}\)

By performing this calculation, we find there are approximately 9.06 moles of oxygen in the tank.
Knowing the number of moles enables us to explore further properties of the gas, such as pressure and volume under different conditions.

The Combined Gas Law is an essential tool for understanding how changes in pressure, volume, and temperature affect a sample of gas. It combines three fundamental gas laws - Boyle's Law, Charles's Law, and Gay-Lussac's Law - into a single relationship:

• \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\)

Here, \(P\), \(V\), and \(T\) represent the pressure, volume, and temperature of the gas, respectively. The subscript '1' refers to the initial state, while '2' refers to the final state. This equation assumes the amount of gas remains constant.
In our exercise, the Combined Gas Law helps us find the new volume of oxygen at a different temperature (from 9°C to 26°C) and pressure (from 10.15 atm to 0.95 atm). By substituting the known values into the equation, we solve for the final volume \(V_2\). This demonstrates how the properties of gases change across different conditions.
Remember, temperature must always be in Kelvin for these calculations. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.

Gas pressure calculation heavily relies on the ideal gas law, which is a fundamental equation that describes the behavior of gases. The equation is given by:

• \(PV = nRT\)

where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
For the scuba diver's tank example, we need to calculate the pressure inside the tank. With 9.06 moles of oxygen, a tank volume of 2.3 L (converted to cubic meters for SI units), and a temperature of 282.15 K, we can rearrange the ideal gas law to solve for pressure:

• \(P = \frac{nRT}{V}\)

Using the constant \(R = 8.314 ext{ J/(mol·K)}\), the calculation provides a pressure of approximately 10.15 atm after converting from Pascals. This highlights the importance of unit conversion, as pressure is commonly expressed in atmospheres in practical applications.
Understanding these calculations allows for better handling of gases under various conditions, ensuring safety and efficiency in applications such as scuba diving.