Tutorial 1 EP 209 - Thermal Physics Question 1 - Thermometers

A constant volume gas thermometer contains a gas whose equation of state is

a P + 2 (v − b) = RT v

and another, of identical construction, contains a diﬀerent gas which obeys the ideal gas law, Pv = RT, where v is the molar molar volum volume. e. The thermom thermomete eters rs are calibr calibrate ated d at the ice and steam points. Show that they will give identical values for a temperature measurement. Question 2 - Isotherms

For any thermodynamic (PVT) system in equilibrium, can isotherms for diﬀerent temperatures intersect? Question Question 3 - Diﬀerent Diﬀerent Temp. Temp. Sc Scales ales

The pressure of an ideal gas kept at constant volume is given by the equation P=AT, where T is the thermodynamic temperature and A is a constant. Let a temperature T* be deﬁned by T* = B ln(CT), ln(CT), where where B and C are consta constant nts. s. The pressu pressure re P is 0.1 atm at the triple triple point of water. The temperature T* is 0 degree at the triple point and T* is 100 degree at the steam point. a) Find the values of A, B and C. b) Find the value of T* when P= 0.15 atm. c) Find the value of P when T* is 50 degree. d) What is the value of T* at absolute zero? e) Sketch a graph of T* vs. the Celsius temperature t for -200 o C < t < 200 20 0o C. [Ans: (a) A = 3.66 x 10−4 atm K−1 , B= 321 degree, C = 3.66 x 10 −3 K−1 , (b) 130 degree, (c) 0.12 atm, (d) tends to -inﬁnity.

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Question 4 - Heating

A block of copper at a pressure of 1 atm (approximately 100 kPa) and a temperature of 5o C is kept at constant volume. If the temperature is raised to 10 o C, what will be the ﬁnal pressure? If the vessel holding the block of copper has a negligibly small thermal expansivity and can withstand a maximum pressure of 100 atm, which is the highest temperature to which the system may be raised? (Note: volume expansion coeﬃcient β is three times the linear expansion coeﬃcient α and isothermal compressibility κ is the reciprocal of the bulk modulus B. For this problem, assume that the volume expansivity and isothermal compressibility remains practically constant within the temperature range of 0 to 20 o C at the values of 4.95 x 10−5 K−1 and 6.17 x 10−12 Pa−1 , respectively.) Question 5 - Idea of Partials

A and B are both functions of the variables x and y, and A/B = C. Show that

∂ (lnB) ∂ (lnA) − ∂y ∂y = ∂ (lnA) x ∂ (lnB) x C

∂x ∂y

∂x

y

−

∂x

y

Question 6 - Wires

The tension in a wire is increased quasi-statically and isothermally from F i to F f . If the length, cross-sectional area and isothermal Youngs modulus of the wire remain constant, show that the work done is L W = ( F f 2 − F i2 ) 2AY Question 7 - Dielectrics

A dielectric has an equation of state P= χEV where χ is a function of temperature only. Show that the work done in an isothermal, quasi-static change of state is given by W =

1 2V χ

(P f 2 − P i2 ) =

Vχ

2

(E f 2 − E i2 )

Question 8 - Assumptions

Steam enters through an inlet at a pressure of 2 x 10 6 Pa into a cylinder ﬁtted with a piston. Initially the piston is to the extreme left. The area of cross-section of the piston is 106−2 m2 and the distance moved is 0.5m. Calculate the work done. What assumption, if any, was needed to arrive at your answer?

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Question 9 - Burning

In the ﬁgure given below, one mole of a mono-atomic ideal gas occupies two chambers of a cylinder partitioned by means of a movable piston. The walls of the cylinder as well as the piston are thermal insulators. Initially equal amounts of gas ﬁll both the chambers at (P 0 , V 0 , T 0 ). A coil is burnt in the left chamber. The gas absorbs heat and expands, pushing the partition to the right without any friction. The process is quasi-static. The gas in the right chamber is compressed until its pressure becomes 32 P 0 . Calculate a) the ﬁnal temperature of the two chambers, b) the work done on the right chamber and c) the heat absorbed by the left chamber.

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Question 10 - Special pump

The cylinder of volume 0.6 m 3 is provided with a piston, the wall of which has a stop-cork. The left part of the cylinder contains 0.01 k-mol of an ideal gas ( γ = 1.4) at a temperature of 300 K; the right chamber is evacuated. Initially the gas occupies a third of the total volume of the cylinder. The walls of the cylinder are adiabatic. The piston is moved quasi-statically so that the volume of the gas is doubled. The valve in the piston is then opened so that the gas ﬁlls the entire volume. Calculate the ﬁnal pressure and temperature of the gas, work done, ∆Q and ∆U.

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