Ch. 18 - Kinetic Theory of Gases

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 18 - Kinetic Theory of Gases

Problem 64

Chapter 18, Problem 64

A sauna has 7.8 m³ of air volume, and the temperature is 85°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18–2.)

Verified step by step guidance

Convert the given temperature from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \). This is necessary because many thermodynamic equations require temperature in Kelvin.

Determine the saturation vapor pressure \( P_{\text{sat}} \) at 85°C using the provided Table 18–2. This value represents the maximum pressure of water vapor that the air can hold at this temperature.

Calculate the partial pressure of water vapor needed for 10% relative humidity using the formula: \( P_{\text{vapor}} = 0.10 \cdot P_{\text{sat}} \). This gives the actual vapor pressure corresponding to 10% relative humidity.

Use the ideal gas law \( PV = nRT \) to calculate the number of moles of water vapor required. Rearrange the equation to solve for \( n \): \( n = \frac{P_{\text{vapor}} V}{RT} \), where \( P_{\text{vapor}} \) is the partial pressure, \( V \) is the volume of the sauna (7.8 m³), \( R \) is the ideal gas constant (8.314 J/(mol·K)), and \( T \) is the temperature in Kelvin.

Convert the number of moles of water vapor \( n \) to mass \( m \) using the molar mass of water (18.015 g/mol). Use the formula: \( m = n \cdot M \), where \( M \) is the molar mass. This gives the mass of water that needs to be evaporated to achieve 10% relative humidity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relative Humidity

Relative humidity is a measure of the current amount of water vapor in the air compared to the maximum amount of water vapor the air can hold at a given temperature. It is expressed as a percentage. For example, at 85°C, air can hold more moisture than at lower temperatures, so increasing the relative humidity from 0% to 10% means adding a specific amount of water vapor to the air.

Specific Humidity

Specific humidity refers to the mass of water vapor present in a unit mass of air, typically expressed in grams of water vapor per kilogram of air. This concept is crucial for calculating how much water needs to be added to achieve a desired relative humidity, as it directly relates to the volume of air and the temperature, which affects the air's capacity to hold moisture.

Evaporation and Latent Heat

Evaporation is the process by which water changes from a liquid to a gas, requiring energy in the form of heat, known as latent heat. This energy is absorbed from the surrounding environment, which can affect the temperature and humidity levels. Understanding the latent heat of vaporization is essential for calculating how much water must evaporate to achieve the desired increase in relative humidity in the sauna.

Recommended video:

Guided course

10:40

Latent Heat & Phase Changes

Related Practice

Textbook Question

A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

views

Textbook Question

At room temperature, it takes approximately 2.45 x 10³ J to evaporate 1.00 g of water. Estimate the average speed of evaporating molecules. What multiple of v_rms (at 20°C) for water molecules is this? (Assume Eq. 18–4 holds.)

views

Textbook Question

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. Can you explain why our atmosphere contains oxygen but not helium?

views

Textbook Question

Estimate the time needed for a glycine molecule (see Table 18–3) to diffuse a distance of 25μm in water at 20°C if its concentration varies over that distance from 1.00 mol/m³ to 0.50 mol/m³. Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u.

Textbook Question

Calculate the total water vapor pressure in the air on the following day: a hot summer day, with the temperature 30°C and the relative humidity at 75%.

views

Textbook Question

A scuba tank has a volume of 3100 cm³. For very deep dives, the tank is filled with 50% (by volume) pure oxygen and 50% pure helium. What is the ratio of the average kinetic energies of the two types of molecule?