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Chapter 10: Problem 12

Using the ideal gas law, show that at STP, the molar volume of an ideal gas is \(22.4 \mathrm{~L}\).

Short Answer

Expert verified
At STP, using P = 1 atm, T = 273.15 K, R = 0.0821 L⋅atm/(mol⋅K), and n = 1 mol, the ideal gas law gives V = (1 mol)(0.0821 L⋅atm/(mol⋅K))(273.15 K) / (1 atm) = 22.4 L.

Step by step solution

01

State the Ideal Gas Law

Firstly, recall the ideal gas law which is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
02

Insert Standard Temperature and Pressure (STP) Values

Standard Temperature and Pressure (STP) is defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. Insert these values into the Ideal Gas Law.
03

Use the Ideal Gas Constant for the Conditions at STP

At STP, the gas constant (R) is typically given in units of L⋅atm/(mol⋅K). The accepted value is 0.0821 L⋅atm/(mol⋅K). Insert this value into the ideal gas law equation.
04

Calculate the Molar Volume

Solve for the volume (V) with n set to 1 mole to find the molar volume of the gas at STP using the equation V = nRT/P.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Volume
The molar volume of a substance is the volume occupied by one mole of that substance. In the context of gases, molar volume is particularly important because it helps us compare how different gases behave under the same conditions. Using the ideal gas law, we can calculate the molar volume at STP (Standard Temperature and Pressure).

At STP, one mole of an ideal gas occupies a volume of approximately \(22.4 \mathrm{~L}\). This value comes from the ideal gas law when the temperature is set to 273.15 K and the pressure is 1 atm. The calculation explicitly demonstrates this molar volume and serves as a reference point for comparing the behavior of real gases under standard conditions.
STP (Standard Temperature and Pressure)
STP is a commonly used standardized set of conditions for measuring gas volume in terms of temperature and pressure. It stipulates a temperature of 273.15 K (0°C) and a pressure of 1 atmosphere (atm). These conditions provide a baseline for scientists to compare the behavior of different gases.

When working with gases under STP, we can use the ideal gas law to predict how much space a certain amount of gas will take up, or how changes in pressure and temperature will affect the gas. STP simplifies calculations as the variables have known and constant values.
Gas Constant (R)
The gas constant (R) is a fundamental parameter that appears in several equations of state, including the ideal gas law. It is crucial because it connects the thermal energy scale to mechanical energy units through the behavior of gases. For calculations at STP, it's commonly used in the value of 0.0821 L⋅atm/(mol⋅K).

Understanding the value and dimensions of R is essential for students as it is key to interpreting and predicting the behavior of gases. Whether you're calculating the volume a gas will occupy at STP or determining the effects of changing conditions, 'R' serves as the bridge between the observable properties of gases (pressure, volume, and temperature) and the amount of gas present (number of moles).

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