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Chapter 4: Problem 3

A liquid is irregularly stirred in a well-insulated container and thereby experiences a rise in temperature. If the system is the liquid: (a) Has heat been transferred? (b) Has work been done? (c) What is the sign of \(\Delta U\) ?

Short Answer

Expert verified
No heat has been transferred. Work has been done. \(\Delta U\) is positive.

Step by step solution

01

Determine if heat has been transferred

Identify whether the liquid, as the system, has exchanged heat with the surroundings. Since the container is well-insulated, no heat can be transferred out of or into the container. Therefore, the answer is no to heat transfer.
02

Check if work has been done on the liquid

Consider the process described: the liquid is stirred irregularly. Stirring involves applying force to the liquid, which means work is done on the liquid.
03

Determine the sign of \(\backslash\backslashDelta U\)

Since the work done on the liquid causes an increase in its temperature, this implies an increase in the internal energy of the liquid. Therefore, \(\backslash\backslashDelta U\) is positive.

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Key Concepts

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

Heat Transfer
Understanding heat transfer is crucial in thermodynamics. Heat transfer occurs when there is energy exchange between a system and its environment due to a temperature difference. In the given problem, the container is well-insulated, which means it's designed to prevent heat exchange. This insulation ensures that no heat enters or leaves the system. Such a condition leads to the analysis that there is no heat transfer in or out of the liquid. In thermodynamic terms, this is denoted as \( q = 0 \). Knowing this helps us focus on other sources of energy change within the system, such as work done.
Work Done
Work done on a system can change its internal energy. When we talk about work in thermodynamics, we're usually discussing the energy transferred to or from a system by forces that cause displacement. In this exercise, the liquid is stirred. Stirring involves applying an external force which creates motion within the liquid. Therefore, we say work (\( W \)) is done on the liquid. This work increases the liquid's energy levels. Understanding the concept of work done helps in computing energy changes in various processes.
Internal Energy
Internal energy represents the total energy contained within a system. This includes kinetic and potential energy at the microscopic level. The symbol \( U \) is used to denote internal energy, and \( \Delta U \) symbolizes the change in internal energy. In this problem, since work is done on the liquid and no heat is transferred, the system's internal energy increases. Thus, we observe a rise in temperature. Mathematically, for an insulated system, \( \Delta U = W \). Here, \( \Delta U \) is positive, indicating an increase in the liquid's internal energy due to the work performed by stirring.

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Most popular questions from this chapter

A cylindrical metal can, blackened on the outside, \(0.10 \mathrm{~m}\) high and \(0.05 \mathrm{~m}\) in diameter, contains liquid \({ }^{4} \mathrm{He}\) at its normal boiling point of \(4.22 \mathrm{~K}\), at which its heat of vaporization is \(20.4 \mathrm{~kJ} / \mathrm{kg}\). Completely surrounding the helium can are walls maintained at the temperature of liquid nitrogen \((77.35 \mathrm{~K})\), and the intervening space is continuously evacuated to a very low pressure. How much helium is lost per hour?

A thick-walled insulated metal chamber contains \(n_{i}\) moles of helium at high pressure \(P_{i}\). It is connected through a valve with a large, almost empty gasholder in which the pressure is maintained at a constant value \(P^{\prime}\), very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the gasholder until the pressure on the two sides of the valve is equalized. Prove that $$ \frac{n_{f}}{n_{i}}=\frac{h^{\prime}-u_{i}}{h^{\prime}-u_{f}} $$ where \(n_{f}=\) number of moles of helium left in the chamber, \(u_{i}=\) initial molar internal energy of helium in the chamber, \(u_{f}=\) final molar internal energy of helium in the chamber, and \(h^{\prime}=u^{\prime}+P^{\prime} v\) (where \(u^{\prime}=\) molar internal energy of helium in the gasholder; \(v^{\prime}=\) molar volume of helium in the gasholder).

The solar constant is the incident energy per unit of time on a unit area of a surface placed at right angles to a sunbeam just outside the earth's atmosphere. The value of the solar constant is \(1.37 \mathrm{~kW} / \mathrm{m}^{2}\). The area of a sphere with radius \(93,000,000\) miles is \(2.79 \times 10^{23} \mathrm{~m}^{2}\), and the surface area of the sun is \(6.09 \times 10^{18} \mathrm{~m}^{2}\). Assuming that the sun is a blackbody, calculate its surface temperature,

The molar heat capacity at constant pressure \(C_{P} / n\) of a gas varies with the temperature according to the equation $$ \frac{C_{P}}{n}=a+b T-\frac{c}{T^{2}} $$ where \(a, b\), and \(c\) are constants. How much heat is transferred during an isobaric process in which \(n\) moles of gas experience a temperature rise from \(T_{i}\) to \(T_{f} ?\)

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