Heat and First Law of Thermodynamics

Heat and First Law of Thermodynamics
Objectives:

Understand the concepts of heat, internal energy, and thermodynamic processes.

Define and discuss the calorie, heat capacity, specific heat, and latent heat.

Provide a qualitative description of different types of phase changes which a substance may undergo, and the changes in energy which accompany such processes.

Discuss the possible mechanisms which can give rise to heat transfer between a system and its surroundings; that is, heat conduction, convection and radiation. You should also be able to state the basic law of heat conduction, and give a realistic example of each heat transfer mechanism.

Understand how work is defined when a system undergoes a change in state, and the fact that work (like heat) depends on the path taken by the system. You should know how to sketch processes on a PV diagram, and calculate work using these diagrams.

State the first law of thermodynamics (D U=Q-W), and explain the meaning of the three forms of energy contained in this statement.

Discuss the implications of the first law of thermodynamics as applied to (a) an isolated system, (b) a cyclic process, (c) an adiabatic process, and (d) an isothermal process.

Calculate the work done when an ideal gas expands during an isothermal process.

Skills

Many applications of the first law of thermodynamics deal with the work done by (or on) a system which undergoes a change in state.

For example, as a gas is taken from a state whose initial pressure and volume are P_i and V_i, and whose final pressure and volume are P_f and V_f, the work can be calculated if the process can be drawn on a PV diagram.

The work done during the expansion is given by the integral expression, which numerically represents the area under the PV curve.

It is important to recognize the work depends on the path taken as the gas goes from i to f. That is, W depends on the specific manner in which the pressure P changes during the process.

Heat and thermal energy

There is a major difference between heat and internal energy.

When two systems at different temperatures are in contact with each other, energy will transfer between them until they reach the same temperature (that is, when they are in equilibrium with each other).

This energy is called heat, or thermal energy, and the term "heat flow" refers to an energy transfer as a consequence of a temperature difference.

On the other hand, internal energy is the energy a substance has at some temperature.

In the next chapter, we shall show that the energy of an ideal gas is associated with the internal motion of its atoms and molecules. In other words, the internal energy of a gas is essentially its kinetic energy on the microscopic scale; the higher temperature of the gas, the greater its internal energy.

Heat capacity and specific heat

The unit of heat is calorie (cal), defined as the amount of heat necessary to increase the temperature of 1 g of water from 14.5^oC to 15.5^oC. (1 cal=4.186 J)

Note that the "Calorie", which is used in describing the energy equivalent of foods, is actually a kilo-calorie.

The heat capacity, C’, of a substance is defined as the amount of heat required to increase the temperature of that substance by one Celsius degree. The unit is cal/^oC or J/^oC.

The specific heat c of a substance of mass m equals its heat capacity per unit mass, i.e., c=C’/m.

The molar heat capacity of a substance is defined as the heat capacity per unit mole, i.e., C=C’/n.

The heat energy Q transferred between a system of mass m and its surroundings for a temperature change D T is given by Q=C’D T=mcD T.

In general, specific heats measured under constant pressure, c_p, are different from those measured under constant volume, c_v. However, the difference for liquids and solids are usually no more than a few per cent and is often neglected.

Since experimental measurements on solids and liquids are easier to perform under the constant-pressure conditions, it is usually c_p that is measured.

Latent heat

A substance may undergo a phase change when heat is transferred between the substance and its surroundings, where the flow of heat does not result in a change in temperature. When a phase change occurs, the physical characteristics of the substance change from one form to another, and can be described in terms of a rearrangement of molecules when heat is added to or removed from a substance. The heat Q required to change the phase of a mass m (called heat of transformation) is given by

Q=mL

where L is called the latent heat. The latent hear of fusion is a parameter used to characterize a solid-to-liquid phase change; the latent heat of vaporization characterizes the liquid-to-gas phase change.

P2.1. A 50-g ice cube at 0° C is heated until 45 g has become water at 100 ° C and 5 g has been converted to steam. How much heat was added to do this ? (For water, the latent heats of fusion and vaporization are 334 and 2260 kJ kg^-1 respectively and the specific heat of water at constant pressure is 4180 J kg^-1K^-1

Work and heat in thermodynamic processes

In the macroscopic approach to thermodynamics, we describe the state of a system with such variables as pressure, volume, temperature, and internal energy.

Q2.4. Pressure, volume, and temperature are variables of state for thermodynamic system. Is heat a variable of state? Explain.
Consider a gas contained in a cylinder fitted with a movable piston. In equilibrium, the gas occupies a volume V and exerts a uniform pressure P on the cylinder walls and piston. If the piston has a cross-sectional area A, the force exerted by the gas on the piston is F=PA. Now let us assume that the gas expands quasi-statically, that is, slowly enough to allow the system to move through an (infinite) series of equilibrium states. As the piston moves up a distance dy, the work done by the gas on the piston is

dW=Fdy=PAdy

Since Ady is the increase in the volume of the gas dV, we can express the work done as

dW=PdV

The work done by a gas which undergoes an expansion or compression from initial volume V_i to final volume V_f is given by

To evaluate this integral, one must know how the pressure varies during the process. Note that a process is not specified merely by giving the initial and final states. That is, a process is a fully specified change in the state of the system. In general, the work done equals the area under the PV curve bounded by V_i and V_f, and the function P.

The pressure is generally not constant, so you must exercise care in evaluating W from this equation. For example, when we study the case for a quasi-static isothermal compression of a liquid/solid, the equation isier. The isothermal compressibility is defin still applicable. We need to write the pressure P as a function of the volume V in order to do the integration.During compression of a liquid/solid, the compressibility k and the volume V change only slightly and so can be treated as constants in the integration.

(a) The system goes from an initial state i to a final state by means of a thermodynamic process. The area marked W represents the work done by the system during this process. The work is positive, because the process proceeds to the right on the graph.

(b) Another process for moving between the two states; the work is now greater than that in (a).

(d) The work can be made as small as you like or as large as you like.

If the gas is compressed, V_f<V_i, and the work is negative. That is, work is done on the gas. If the gas expands, V_f>V_i, the work is positive, and the gas does work on the piston. If the gas expands at constant pressure, called an isobaric process, W=P(V_f-V_i).

(e) When the volume is reduced (by some external force), the work done by the system is negative.

(f) The net work done by the system during a (closed) cycle is represented by the enclosed area, which is the difference in the areas beneath the two curves that make up the cycle.

The first law of thermodynamics

The first law of thermodynamics is a generalization of the law of conservation of energy that includes possible changes in internal energy. The first law of thermodynamics also provides us with a connection between the microscopic and macroscopic worlds. We have seen that energy can be transferred between a system and its surroundings in two ways. One is work done by or on the system. This mode of energy exchange results in measurable changes in the macroscopic variables of the system, such as the pressure, temperature and volume of a gas. The other is heat transfer, which takes place at the microscopic level.

It states that the change in internal energy of a system, D U, is given by

D U = D Q-W ,

which is shown as

where D Q is the heat added to the system and W is the work done by the system. Note that by convention, D Q is positive when hear enters the system and negative when heat is removed from the system. Likewise, W can be positive or negative as mentioned earlier.

EXAMPLE

The drinking bird (AKA "Dippy Bird") illustrates the conversion of thermal energy into mechanical energy. The bird consists of two hollow glass chambers (head and bottom) which are coupled by a double walled glass tube. The outer tube provides mechanical support, and the inner tube extends into the bottom chamber below the surface of a colored fluid with suitably high vapor pressure. The head of the bird is coated with a fuzzy material, and is initially soaked in water so that it will begin to cool by evaporation. This provides the temperature difference from head to tail necessary to run the heat engine. As the head cools, the colored fluid is observed to rise up from the bottom of the bird through the neck, gradually shifting the center of gravity of the bird toward its head. The bird bends at the hips and dips its bill into a glass of water (thus keeping the head wet and cooler than the tail). As the fluid continues to rise into the head, the fluid level in the bottom of the bird eventually drops below the end of the connecting tube. This allows vapor to be pulled up through the neck to equilibrate the pressure. The fluid runs back down into the bottom of the bird, the bird stands up again, and the cycle repeats indefinitely.

PROCESS OUTLINE IN A CYCLE

Applying the rule to the power plant shown in figure above gives,

Q = Qin - Qout
W = Win - Wout
Qin + Win - Qout - Wout = 0

where,
Qin = Heat supplied to the system through boiler,
Win = Feed-pump work,
Qout = Heat rejected from the system by condenser,
Wout = Turbine work.

INTERNAL COMBUSTION ENGINE

P2.8. A thermodynamic system undergoes a process in which its internal energy decreases by 300 J. If at the same time, 120 J of work is done on the system, find the heat transferred to or from the system.

P2.9. A vertical piston-cylinder assembly contains a gas which is compressed by a frictionless piston weighing 3 kN. During an interval of time, a paddle-wheel within the cylinder does 6800 Nm of work on the gas. If the heat transfer out of the gas is 8.7 kJ and the change in the internal energy is -1 kJ, determine the distance the piston moves. The area of the piston is 5 x 10^-3 m², and the atmospheric pressure acting on the outside of the piston is 95 kPa.

The initial and final states must be equilibrium states; however, the intermediate states are, in general, non-equilibrium states since the thermodynamic coordinates undergo finite changes during the thermodynamic process.

Although D Q and W both depend on the path, the quantity D Q-W, that is,

the change in internal energy, is independent of the path.

For an infinitesimal change of the system, we can express the first law of thermodynamics in the form of

dU = dQ - dW

In fact, dQ and dW are not exact differentials since both Q and W are not functions of the system’s coordinates. That is, both Q and W depend on the path taken between the initial and final equilibrium states, during which time the system interacts with its environment. On the other hand, dU is an exact differential and the internal energy U is a state variable.

An isolated system is one which does not interact with its surroundings. In such a system, Q=W=0, so it follows from the first law that D U=0. That is, the internal energy of an isolated system cannot change.

A cyclic process is one that originates and ends up at the same state. In this situation, D U=0, so from the first law we see that Q=W. That is, the work done per cycle equals the heat added to the system per cycle. This is important to remember when dealing with heat engines in the next chapter.

Q2.7. Must the internal energy of a system increase if heat is added to the system?

Some applications of the first law of thermodynamics

An adiabatic process is a process in which no heat enters or leaves the system, that is, Q=0. The first law applied to this process gives D U=-W. A system may undergo an adiabatic process if it is thermally insulated from its surroundings.

An isobaric process is a process which occurs at constant pressure. For such a process, the heat transferred and the work done are non-zero.

P2.10. Gas in a container is at a pressure of 200 kPa and a volume of 3 m³what is the work done by the gas if (a) it expands at constant pressure to twice its initial volume and (b) it is compressed at constant pressure to one third its initial volume ?

*P2.11. An ideal gas initially at 300 K undergoes an isobaric expansion at pressure of 25 Nm^-2. If the volume increases from 1 m³ to 3 m³ and 80 J of heat is added to the gas, find (a) the change in internal energy of the gas and (b) its final temperature.

An isovolumetric process is one which occurs at constant volume. By definition, W=0 for such as process (since dV=0), so from the first law it follows that D U=Q. That is, all of the heat added to the system kept at constant volume goes into increasing the internal energy of the system.

An isothermal process is one which occurs at constant temperature. During such a process, the change in internal energy results from both heat transfer and work done. The work done during the isothermnal expansion of an ideal gas can be calculated from

W = nRTln(Vf/Vi).