Problem 88 Using the ideal gas law, \(P V=n... [FREE SOLUTION] (2024)

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Understanding how to calculate the volume of a gas is a fundamental concept in chemistry, particularly when dealing with reactions and processes involving gases. The volume of a gas can be predicted using the ideal gas law, expressed as the formula: \(PV = nRT\).

To calculate the gas volume, this relationship requires us to know the number of moles of the gas (\(n\)), the temperature (\(T\)) in Kelvin, the pressure (\(P\)) exerted by the gas, and the ideal gas constant (\(R\)). For example, in the given exercise where we have 0.510 moles of hydrogen gas (\(H_2\)) at a temperature of 320.15 K and a pressure of 1.6 atm, the volume (\(V\)) can be calculated by rearranging the formula to solve for \(V\): \(V = \frac{nRT}{P}\).

Once you plug the known values into the equation, the calculation yields the volume of the hydrogen gas under the specified conditions. This method is widely used in laboratory settings and industrial applications to determine the required volume of a gas for a particular reaction or process. It is also a stepping stone toward more complex gas-related calculations.

The molar mass of a substance is the mass of one mole of that substance, usually expressed in grams per mole (g/mol). It plays a crucial role in chemistry when converting between the mass of a substance and the number of moles. For gases, the ideal gas law provides a handy way to determine molar mass if other properties of the gas are known.

In the context of the ideal gas law, the molar mass (\(M\)) can be calculated by rearranging the density form of the law: \(d = \frac{MP}{RT}\), which gives us \(M = \frac{dRT}{P}\). Here, density (\(d\)) is mass per unit volume. This concept is beautifully illustrated in part (d) of the exercise, where the unknown gas has a density of 2.58 g/L at a temperature of 300.15 K and a pressure of 1.00 atm. By rearranging the equation and substituting the known values, we can solve for the molar mass of the gas.

To illustrate, imagine a balloon filled with an unknown gas. By measuring the mass and volume of the gas in the balloon, while knowing the temperature and pressure, one can calculate the molar mass and potentially identify the gas. Molar mass determination is thus essential for chemical identification and reaction stoichiometry.

Gas density is a measure of a gas's mass per unit volume and is an important property in both scientific and industrial applications. Understanding how to compute the gas density involves manipulating the ideal gas law equation appropriately.

Starting with the ideal gas law, \(PV = nRT\), and knowing that the number of moles (\(n\)) is equal to mass (\(m\)) divided by the molar mass (\(M\)), we can rewrite the equation in terms of mass and volume. This gives us the density equation: \(d = \frac{PM}{RT}\), where \(d\) is density, \(P\) is pressure, \(M\) is molar mass, \(R\) is the ideal gas constant, and \(T\) is temperature. Knowing any three of these properties allows us to calculate the fourth.

For instance, as performed in part (c) of the exercise, the density of carbon dioxide (\(CO_2\)) at 4.00 atm pressure and 293.15 K temperature can be computed by substituting the known values into the density equation. In practical scenarios, this computation enables us to design appropriate storage systems for gases and helps in determining the buoyancy and lifting properties of gases, which is vital for applications like hot air balloons and even predicting the atmosphere's behavior.