Question

If 4 moles of an ideal gas at 300 K occupy volume of \(89.6 \mathrm{~L}\), then pressure of the gas will be (a) \(2 \mathrm{~atm}\) (b) \(1 \mathrm{~atm}\) (c) \(1.099 \mathrm{~atm}\) (d) \(2.910 \mathrm{~atm}\)

Short answer

The pressure of the gas is approximately 1.099 atm, so the correct answer is (c) 1.099 atm.

Step-by-step solution

Recall the Ideal Gas Law

The ideal gas law can be stated as 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.

Identify the given values

For this problem, we are given the following: n = 4 moles (amount of the gas), V = 89.6 L (volume occupied by the gas), and T = 300 K (temperature). We are also given that R (ideal gas constant) in units of L.atm/(K.mol) is approximately 0.0821.

Solve for Pressure (P)

Using the ideal gas law equation, we can solve for P by rearranging it to P = nRT/V. Substituting in the known values: P = (4 moles * 0.0821 L.atm/(K.mol) * 300 K) / 89.6 L.

Perform the calculation

Carrying out the multiplication and division to find P gives: P = (4 * 0.0821 * 300) / 89.6 = 98.52 / 89.6 = 1.099 atm (approx).

Select the correct answer

The calculated pressure is approximately 1.099 atm, which corresponds to option (c).

Key concepts

Chemistry Problem Solving

Understanding how to solve chemistry problems involves recognizing the problem type and recalling the relevant formulas. For gas-related problems, it typically means identifying the variables involved such as pressure, volume, temperature, and the amount of gas in moles.

In our example, the problem begins by providing us with three known values and asks us to solve for the fourth using the Ideal Gas Law. To tackle this problem systematically, we follow a series of steps which include identifying the knowns and unknowns, recalling the correct equation, substituting in the values, and solving for the unknown variable.

This structured approach not only simplifies the problem but also reduces chances for error by organizing the information before performing the calculations.

Gas Laws

Gas laws are fundamental to understanding the behaviour of gases under different conditions. These laws help us predict how a gas will change when we modify pressure, volume, and temperature.

The Ideal Gas Law, Boyle’s Law, Charles’s Law, and Avogadro’s Law are among the key laws governing gaseous substances. The Ideal Gas Law unifies the other three, allowing us to solve for any one of the four variables (P, V, n, T) if we know the other three.

It's essential to understand that these laws are based on ideal conditions where gases do not have intermolecular forces and occupy no volume, which is a close approximation for real gases under low pressure and high temperature.

PV=nRT

Understanding the Equation

The equation PV=nRT is known as the Ideal Gas Law. Each letter represents a physical quantity: P is pressure, V is volume, n is the number of moles (quantity of substance), R is the ideal gas constant, and T is the absolute temperature in Kelvin.

This equation is powerful as it relates the four state variables of a gas which are usually dependent on one another. If we manipulate any of these variables, we can predict how the other variables will be affected. For instance, increasing the temperature while keeping the volume constant will also increase the pressure, as shown in the equation.

Molar Volume

The Volume of One Mole of Gas

Molar volume is the volume occupied by one mole of any gas at a specified temperature and pressure. At standard temperature and pressure (0°C and 1 atm), the molar volume of an ideal gas is approximately 22.4 L.

It’s important to note that molar volume is an indirect application of Avogadro's Law, which states that equal volumes of all gases at the same temperature and pressure contain the same number of molecules, regardless of the chemical nature of the gas.

Thus, by knowing the molar volume, we can make conversion between the amount of gas in moles and its volume, and this can be further related to the Ideal Gas Law equation for solving various problems involving gases.