Textbook Question
How many moles of water are there in 1.00 L at STP? How many molecules?
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20. Heat and Temperature
Moles and Avogadro's Number
4:48 minutesProblem 52
Textbook Question
Compare the value for the density of water vapor at exactly 100°C and 1 atm (Table 13–1) with the value predicted from the ideal gas law. Why would you expect a difference?
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Step 1: Recall the density formula for a gas, which is given by \( \rho = \frac{m}{V} \), where \( m \) is the mass of the gas and \( V \) is its volume. For water vapor at 100°C and 1 atm, the density can also be calculated using the ideal gas law.
Step 2: Write the ideal gas law equation: \( PV = nRT \). Here, \( P \) 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. Rearrange this equation to express \( V \) in terms of \( n \): \( V = \frac{nRT}{P} \).
Step 3: Substitute \( n = \frac{m}{M} \) (where \( M \) is the molar mass of water vapor) into the equation for \( V \). This gives \( V = \frac{mRT}{MP} \). Now, substitute this expression for \( V \) into the density formula \( \rho = \frac{m}{V} \), resulting in \( \rho = \frac{MP}{RT} \).
Step 4: Use the known values for water vapor: \( M = 18.015 \ \text{g/mol} \), \( P = 1 \ \text{atm} \), \( R = 0.0821 \ \text{L·atm/(mol·K)} \), and \( T = 373.15 \ \text{K} \) (100°C in Kelvin). Substitute these values into the formula \( \rho = \frac{MP}{RT} \) to calculate the density of water vapor as predicted by the ideal gas law.
Step 5: Compare the calculated density from the ideal gas law with the tabulated value for the density of water vapor at 100°C and 1 atm (from Table 13–1). Discuss why the values differ: the ideal gas law assumes no intermolecular forces and that the gas particles occupy no volume, which is not entirely accurate for real gases like water vapor, especially near the boiling point where intermolecular forces are significant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Density of Water Vapor
The density of water vapor at a specific temperature and pressure is a measure of how much mass of water vapor is contained in a given volume. At 100°C and 1 atm, water vapor behaves differently than a liquid, and its density can be influenced by intermolecular forces and the phase of the substance. This density can be found in tables or calculated using the ideal gas law.
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Ideal Gas Law
The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law assumes that gas particles do not interact and occupy no volume, which can lead to discrepancies when applied to real gases under certain conditions.
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Real Gas Behavior
Real gases deviate from ideal behavior under high pressure and low temperature, where intermolecular forces and the volume of gas particles become significant. At 100°C and 1 atm, water vapor may exhibit such deviations due to hydrogen bonding and the presence of liquid water in equilibrium. This can result in a measured density that differs from the value predicted by the ideal gas law, highlighting the limitations of the ideal gas approximation.
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