Chapter 6: Problem 33
What is the molar volume of an ideal gas at (a) \(25^{\circ} \mathrm{C}\) and 1.00 atm; \((b) 100^{\circ} \mathrm{C}\) and 748 Torr?
Short Answer
Expert verified
The molar volume of an ideal gas is (a) \(24.47 L/mol\) at \(25^{\circ} \mathrm{C}\) and 1.00 atm and (b) \(31.24 L/mol\) at \(100^{\circ} \mathrm{C}\) and 748 Torr.
Step by step solution
01
Convert Temperature to Kelvin
First, the given temperatures need to be converted from degrees Celsius to Kelvin using the formula \(K = C + 273.15\). For (a) the temperature in Kelvin is \(25 + 273.15 = 298.15 K\). For (b) the temperature in Kelvin is \(100 + 273.15 = 373.15 K\).
02
Convert Pressure
Next, the pressure of each scenario must be in the same units as the ideal gas constant. For (a) the pressure is already in atm so it stays as 1.00 atm. For (b), the pressure must be converted from Torr to atm using the conversion factor \(1 atm = 760 Torr\), so the pressure in atm is \(748 Torr *(1 atm / 760 Torr) = 0.984 atm\).
03
Calculate Molar Volume
Now we can solve for V in the ideal gas law equation, rearranged to \(V = nRT / P\). For (a), the molar volume is \((1 mol * 0.0821 L*atm / (mol*K) * 298.15 K) / 1.00 atm = 24.47 L/mol\). For (b), the molar volume is \((1 mol * 0.0821 L*atm / (mol*K) * 373.15K) / 0.984 atm = 31.24 L/mol\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Molar Volume
The concept of molar volume refers to the volume occupied by one mole of a substance. Specifically for gases, the molar volume combines ideas from the Ideal Gas Law to show how much space a mole of gas takes up under certain conditions. For an ideal gas, molar volume can be calculated using the equation \[ V = \frac{nRT}{P} \]In this equation:
- \( V \) is the volume occupied by the gas in liters,
- \( n \) is the number of moles (typically 1 mole),
- \( R \) is the ideal gas constant \(0.0821 \text{ L*atm / (mol*K)}\),
- \( T \) is the temperature in Kelvin,
- \( P \) is the pressure in atm.
Temperature Conversion
When dealing with gas laws, it's crucial to convert temperatures to Kelvin from Celsius. Kelvin is the SI unit of temperature and is essential because it starts at absolute zero, eliminating negative temperatures. The conversion process is simple: \[ K = C + 273.15 \]This formula accounts for the Celsius temperature given and adds 273.15 to convert it into Kelvin. For example, a temperature of \(25^{\circ} C\) becomes \(298.15 K\), and \(100^{\circ} C\) becomes \(373.15 K\). The Kelvin scale is important for scientific calculations, as it offers a true measure of thermal energy.
Pressure Conversion
Pressure conversions are necessary when handling equations in the Ideal Gas Law. Not all pressures directly match the units needed for the gas constant \( R \), which uses atm. To convert pressure from Torr to atm, you apply the known relationship: \[ 1 \text{ atm} = 760 \text{ Torr} \]For example, if the pressure is \(748 \text{ Torr}\), you can convert it by:\[ 748 \text{ Torr} \times \frac{1 \text{ atm}}{760 \text{ Torr}} = 0.984 \text{ atm} \]By converting pressures into a common unit, calculations with gases become accurate and clear, leading to meaningful results.
Kelvin
Kelvin is the standard unit of temperature measurement used in scientific calculations. It represents an absolute scale starting at absolute zero, where no thermal energy is present. This unit is advantageous because it ensures all thermodynamic equations are linear, avoiding negative values that can complicate calculations. To convert temperatures from Celsius to Kelvin, the formula is used:\[ K = C + 273.15 \]Kelvin is pivotal when applying the Ideal Gas Law, as it provides the temperature measure needed to maintain consistency and accuracy in scientific assessments.
Torr
Torr is another unit of pressure that is often used interchangeably with mmHg. Named after the Italian scientist Evangelista Torricelli, one Torr is defined as the pressure exerted by a 1 mm column of mercury at sea level. While Torr is useful, most ideal gas calculations require pressure in atmospheres (atm) to match the units of the ideal gas constant. Therefore, converting Torr to atm is a common step in these calculations. This is done using:\[ 1 \text{ atm} = 760 \text{ Torr} \]Understanding Torr and how it fits into larger calculations helps in interpreting and converting pressures to ensure consistency and accuracy in solving gas law problems.