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Show that 1.00 mol of ideal gas under STP occupies a volume of \(0.0224 \mathrm{~m}^{3}=22.4 \mathrm{~L}\).

Short Answer

Expert verified
1.00 mol of ideal gas at STP occupies 22.4 L.

Step by step solution

01

Identify Given Values and Conditions

The problem states that we have 1.00 mol of an ideal gas under STP conditions. Standard Temperature and Pressure (STP) are defined as a temperature of 0掳C or 273 K and a pressure of 1 atm (about 1.01 脳 10^5 Pa).
02

Use the Ideal Gas Law

The Ideal Gas Law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature. At STP, \( R = 8.314 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1} \) or 0.0821 L路atm路mol鈦宦孤稫鈦宦.
03

Substitute Known Values into the Equation

Use \( P = 101325 \) Pa, \( n = 1.00 \) mol, \( R = 8.314 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1} \), and \( T = 273 \) K. Substitute these into the ideal gas law: \( V = \frac{nRT}{P} \).
04

Calculate the Volume

Insert the known values into the formula: \( V = \frac{(1.00 \text{ mol})(8.314 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1})(273 \text{ K})}{101325 \text{ Pa}} \). Simplify the expression to find \( V \approx 0.0224 \mathrm{~m}^{3} \). Convert cubic meters to liters: \( 0.0224 \mathrm{~m}^{3} = 22.4 \mathrm{~L} \).
05

Verify the Result

The computed volume, 22.4 L, matches the expected volume for 1.00 mol of an ideal gas at STP, confirming the calculation is correct.

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Key Concepts

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

Standard Temperature and Pressure (STP)
Standard Temperature and Pressure, abbreviated as STP, is a reference point in the study of gases. It is important because it provides a set of conditions to compare the behavior of gases under controlled environments.
At STP, the temperature is fixed at 0掳C (which corresponds to 273 Kelvin) and the pressure is set to 1 atmosphere (1 atm), equivalent to about 101325 Pascals (Pa).
These conditions are often used in scientific calculations because they set standard benchmarks that allow scientists and students to predict gas behaviors more accurately. When you hear about the volume of 22.4 liters for 1 mole of an ideal gas at STP, it's derived from these specific conditions using the Ideal Gas Law. STP is a hypothetical scenario based on ideal gas assumptions and serves as a beneficial tool in theoretical calculations.
Moles and Avogadro's number
In chemistry, the concept of moles is central for quantitative analysis. A mole is a unit that measures the amount of a substance. It simplifies calculations involving atoms and molecules, which are typically too small or numerous to count individually.
The mole is closely tied to Avogadro's number, which is approximately \(6.022 \times 10^{23}\), defining the number of atoms or molecules in one mole of a substance.
Avogadro's number helps convert between the mass of a substance and the amount of particles it contains, making it a cornerstone concept for understanding chemical reactions and the Ideal Gas Law. In the context of the Ideal Gas Law and this specific problem, when we use 1 mole of gas, it refers to \(6.022 \times 10^{23}\) molecules of that gas, occupying a volume of 22.4 liters at STP.
Gas constant R
The gas constant, symbolized as \(R\), is a fundamental constant in the Ideal Gas Law, expressed as \(PV = nRT\). It bridges the gap between macroscopic and molecular properties, serving as the proportionality constant in the equation.
There are different values for \(R\) depending on the units used:
  • 8.314 J路mol鈦宦孤稫鈦宦 when dealing with units of pressure in Pascals and volume in cubic meters.
  • 0.0821 L路atm路mol鈦宦孤稫鈦宦 when using atmospheres and liters.
In the provided solution, the specific value of \(R = 8.314\) J路mol鈦宦孤稫鈦宦 is used. It is necessary to use consistent units throughout the calculation. This constant is essential for calculating the volume, pressure, or temperature of a gas sample, allowing for accurate predictions under various conditions.

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Most popular questions from this chapter

The highest and lowest recorded air temperatures in the United States are, respectively, \(134^{\circ} \mathrm{F}\) (Death Valley, California, 1913 ) and \(-80^{\circ} \mathrm{F}\) (Prospect Creek, Alaska, 1971). What are these temperatures on the Celsius scale?

A husband buys a helium-filled anniversary balloon for his wife. The balloon has a volume of \(3.5 \mathrm{~L}\) in the warm store at \(74^{\circ} \mathrm{F}\). When he takes it outside, where the temperature is \(48^{\circ} \mathrm{F}\), he finds it has shrunk. By how much has the volume decreased?

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Convert the following to Fahrenheit readings: (a) \(120^{\circ} \mathrm{C}\) (b) \(12^{\circ} \mathrm{C}\) and \((\mathrm{c})-5^{\circ} \mathrm{C}\).

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