Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 65a

Chapter 19, Problem 65a

14 g of nitrogen gas at STP are pressurized in an isochoric process to a pressure of 20 atm. What are the final temperature?

Verified step by step guidance

Step 1: Understand the problem. The process described is isochoric, meaning the volume remains constant. The relationship between pressure and temperature for a gas undergoing an isochoric process is given by the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \( P_1 \) and \( T_1 \) are the initial pressure and temperature, and \( P_2 \) and \( T_2 \) are the final pressure and temperature.

Step 2: Identify the given values. The initial pressure \( P_1 \) is 1 atm (since the gas is at STP), the initial temperature \( T_1 \) is 273.15 K (standard temperature), and the final pressure \( P_2 \) is 20 atm. The final temperature \( T_2 \) is what we need to find.

Step 3: Rearrange the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) to solve for \( T_2 \). This gives \( T_2 = T_1 \cdot \frac{P_2}{P_1} \).

Step 4: Substitute the known values into the equation. Use \( T_1 = 273.15 \, \text{K} \), \( P_1 = 1 \, \text{atm} \), and \( P_2 = 20 \, \text{atm} \). The equation becomes \( T_2 = 273.15 \cdot \frac{20}{1} \).

Step 5: Perform the calculation to find \( T_2 \). This will give the final temperature in Kelvin. Remember to ensure units are consistent throughout the calculation.

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Key Concepts

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

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding gas behavior under various conditions, including changes in pressure and temperature. In this scenario, it will help determine the final temperature of nitrogen gas after being pressurized.

Isochoric Process

An isochoric process is a thermodynamic process in which the volume remains constant. During this type of process, any change in pressure will directly affect the temperature of the gas, as described by the Ideal Gas Law. Understanding this concept is crucial for analyzing how the temperature of nitrogen gas changes when its pressure is increased while keeping the volume fixed.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is defined as a temperature of 0 degrees Celsius (273.15 K) and a pressure of 1 atm. It serves as a reference point for gas calculations and allows for the comparison of gas behaviors under standard conditions. Knowing the initial conditions at STP is essential for calculating the final state of the gas after the isochoric process.

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Textbook Question

Most stars are main-sequence stars, a group of stars for which size, mass, surface temperature, and radiated power are closely related. The sun, for instance, is a yellow main-sequence star with a surface temperature of 5800 K. For a main-sequence star whose mass M is more than twice that of the sun, the total radiated power, relative to the sun, is approximately P/P_sun=1.5(M/M_sun)^3.5. The star Regulus A is a bluish main-sequence star with mass 3.8M_sun and radius 3.1R_sun. What is the surface temperature of Regulus A?

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Textbook Question

14 g of nitrogen gas at STP are adiabatically compressed to a pressure of 20 atm. What is the compression ratio V_max/V_min?

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Textbook Question

Liquid helium, with a boiling point of 4.2 K, is used in ultralow-temperature experiments and also for cooling the superconducting magnets used in MRI imaging in medicine. Storing liquid helium so far below room temperature is a challenge because even a small 'heat leak' will boil the helium away. A standard helium dewar, shown in FIGURE P19.67, has an inner stainless-steel cylinder filled with liquid helium surrounded by an outer cylindrical shell filled with liquid nitrogen at –196°C. The space between is a vacuum. The small structural supports have very low thermal conductivity, so you can assume that radiation is the only heat transfer between the helium and its surroundings. Suppose the helium cylinder is 16 cm in diameter and 30 cm tall and that all walls have an emissivity of 0.25. The density of liquid helium is 125 kg/m³ and its heat of vaporization is 2.1×10⁴ J/kg. What is the mass of helium in the filled cylinder?

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Textbook Question

FIGURE P19.62 shows a thermodynamic process followed by 120 mg of helium. How much heat energy is transferred to or from the gas during each of the three segments?

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Textbook Question

A monatomic gas is adiabatically compressed to 1/8 of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? The molar specific heat at constant volume.

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Textbook Question

FIGURE P19.62 shows a thermodynamic process followed by 120 mg of helium. How much work is done on the gas during each of the three segments?

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