8 Temperature and Ideal Gases

Question 1

Define Temperature

Answer 1

Temperature is the measure of the average microscopic kinetic energy possessed by atoms or molecules of a substance.

Question 2

Define Heat.

Answer 2

Heat is the transfer of energy from one object to another due to a temperature difference between them. (Heat is always transferred from an object with a higher temperature to an object with a lower temperature)

Question 3

State the ways in which heat can be transferred.

Answer 3

Conduction (via collision of particles)
Convection (via movement of particles in a fluid)
Radiation

Question 4

State what it means for two objects to be in thermal contact.

Answer 4

It means that heat can be exchanged between the two objects.

When the hotter object transfers heat to the cooler object until the net heat transfer between them is zero, thermal equilibrium is reached (objects are of the same temperature). (There is always heat transfer from a cooler object as long as its temperature is above 0. However, the rate of heat transfer from the hotter object to the cooler object will be faster than the rate of heat transfer from the cooler object to the hotter object. Therefore, there is a net heat transfer from the hotter to the colder object.)

Question 5

State the zeroth law of thermodynamics.

Answer 5

It states that if objects A and B are each separately in thermal equilibrium with a third object C, then A and B are also in thermal equilibrium with each other.

Question 6

Define thermal equilibrium. Explain when thermal equilibrium is reached.

Answer 6

When there is no net flow of heat between two objects in thermal contact, the two objects are in a state of thermal equilibrium.

When the hotter object transfers heat to the cooler object until the net heat transfer between them is zero, thermal equilibrium is reached (objects are of the same temperature).

Question 7

Can heat be transferred from a colder object to a hotter object?

Answer 7

(There is always heat transfer from a cooler object as long as its temperature is above 0. However, the rate of heat transfer from the hotter object to the cooler object will be faster than the rate of heat transfer from the cooler object to the hotter object. Therefore, there is a net heat transfer from the hotter to the colder object.)

Question 8

Describe what thermometric properties are. State the various criteria that a suitable thermometric property should satisfy.

Answer 8

Thermometric properties are physical properties of a substance that varies with temperature.

A suitable thermometric property should:

1) vary continuously and uniquely with temperature (different values of the thermometric property for different temperatures)

2) the change in the thermometric property must be large enough to enable accurate measurements of temperature. If not, the sensitivity of the thermometer is said to be low.
3) the value of the thermometric property at any temperature within its working range must be reproducible

Question 9

State the thermometric properties in an (a) Liquid-in-glass thermometer, (b) Platinum resistance thermometer, (c) Thermocouple thermometer and (d) Constant volume gas thermometer.

Answer 9

(a) liquid-in-glass thermometer: length of a fixed mass of liquid in a uniform capillary tube
(b) Platinum resistance thermometer: resistance of a platinum wire
(c) Thermocouple thermometer: e.m.f. produced between junctions of dissimilar metals that are at different temperatures
(d) Constant volume gas thermometer: the pressure of a fixed mass of gas at constant volume

Question 10

Differentiate between sensitivity and responsiveness of a thermometer.

Answer 10

The sensitivity of a thermometer tells us how much the quantity changes per unit temperature change while responsiveness is how quickly the quantity changes to a change in temperature.

Question 11

Describe how the empirical celsius scale is derived.

State the equation to calculate an unknown temperature from the derived scale.

Answer 11

Pg 5 and 6 of notes

Question 12

Explain whether different temperature scales are in full agreement with one another.

Answer 12

The choice of different thermometric substances and thermometric properties would lead to a different scale. Agreement between scales occur only at the two fixed points (ice point and steam point). This happens because thermometric properties may not vary linearly with temperature.

Question 13

Explain why experiments using gas thermometers show that thermometer readings are nearly independent of the type of gas used particularly when the pressure is low.

Answer 13

This is because the lower the pressure of a real gas, the closer it approaches a linear relationship between the variation of pressure or volume with temperature, as the behaviour of real gases approaches that of an ideal gas.

Question 14

Define the absolute scale of temperature or the thermodynamic scale of temperature.

Answer 14

The absolute scale of temperature or the thermodynamic scale of temperature is one that is independent of the property of any substance and has an absolute zero. (It is also known as the Kelvin scale where the temperature is symbolised by T and its unit is kelvin K.)

Question 15

Define absolute zero.

Answer 15

Absolute zero is defined as the zero point (0K) of the thermodynamic temperature scale and the temperature at which all substances have minimum internal energy. (It is also the lower fixed point of the thermodynamic temperature scale)

Question 16

State the relationship between the Celsius scale and the Thermodynamic scale (or Absolute temperature scale).

Answer 16

T/K = degree/(0C) + 273.15

Refer to Pg 9 of notes**

Question 17

What is the triple point of water?

Answer 17

When the absolute zero was adopted as the lower fixed point on the thermodynamic temperature scale, the upper fixed point was taken as the triple point of water, which is the single temperature and pressure at which water, water vapour and ice can co-exist in equilibrium in an enclosed cell. The triple point of water occurs at a temperature of 0.01 degree Celsius and us therefore assigned a value of 273.16K on the thermodynamic scale at a pressure of 611.2 Pa.

Question 18

Show how an unknown temperature T can be determined from the constant volume gas thermometer.

Answer 18

Using the constant volume gas thermometer to measure the pressures p at an unknown temperature T and ptr at triple point, T can be determined from T = 273.16 (p/ptr). (Instead of 100 degree Celsius, it is now 273.16 which is the difference between zero/lower fixed point and triple point of water and the triple point of water.)

Question 19

Define an ideal gas.

Answer 19

An ideal gas js a gas which obeys the equation of state pV = nRT at all pressures, volumes and temperatures (where p is the pressure of the gas, V is the volume of the gas, n is the amount if gas in moles, R is the molar gas constant and T is the thermodynamic temperature in K of the gas; p, V and T are state variables that characterise the state if a fixed amount of the system).

Question 20

State Avogadro’s constant.

Answer 20

Avogadro’s constant (NA) is defined as exactly 6.02 × 10^23 particles per mole, NA = 6.02 × 10^23 mol ^(-1).

Question 21

Define the mole.

Answer 21

The mole is defined as the amount of matter containing the number if particles (atoms, ions or molecules) equal to exactly 6.02 × 10^23 particles. (It establishes a connection between a macroscopic amount of matter with the number if microscopic particles within it.)

Question 22

State the alternative expressions of the ideal gas equation.

Answer 22

For 1 mole, pVm = RT where Vm = V/n is the molar volume.

For N molecules, pV = (N/NA)RT where NA is the Avogadro’s constant.

pV = (N/NA)RT -> pV = NkT where k = R/NA is the Boltzmann constant with a value of k = 1.38 × 10 ^(-23) JK ^(-1)

Question 23

State the basic assumptions of the kinetic theory of gases.

Answer 23

1) The gas consists of a large number of molecules in random motion (so that only average behavior is considered).
2) The molecules exert no intermolecular forces on one another except during collisions. (Molecules are relatively far apart and interact only very weakly. They exert negligible forces of attraction on one another. Hence, microscopic kinetic energy = 0 as the forces between the molecules, known as van der Waals forces, decreases rapidly with separation.)
3) Collisions between the molecules and the container are perfectly elastic (no loss in K.E.)
4) The duration of an intermolecular collision is negligible compared with the duration between collisions.
5) The volume of the molecules is negligible compared with the volume occupied by the gas.

Question 24

Explain how molecular movement causes the pressure exerted by a gas and hence, derive the relationship pV = 1/3 (Nm)c^2

Answer 24

If an individual molecule collides with a wall, as shown in the animation, its momentum gets doubled.A gas molecule can move in any direction at a given time - in the x-direction, y-direction or z-direction

Let’s consider the motion in the x-direction, as shown in the animation; let the velocity be cx
If the length of the cube, mass of the molecule and velocity are d, m and v respectively,
Momentum in the x-direction = mcx
Momentum in the -x-direction = -mcx
Change in momentum = -mcx-mcx = 2mUx
Total time taken - from one end to the other and vice versa - = 2d/cx
Rate of change in momentum = -2mcx/(2d/cx)
= -mcx^2/d

According to Newton’s Second Law, the rate of change of momentum is the average force exerted by the molecule on the wall. Therefore, F on wall = F on molecule = mcx^2/d

By considering all the N number of molecules, the total force acting on the wall F total is given by F tot = Nm/d where is the mean-square-speed of all the molecules in the x-direction.

Since pressure, P = force / area
Pressure on the wall, P = [Nm/d] / d^2
= Nm/V where V is the volume of the container, THE cube.

Since velocity in each direction is equal,
c^2 = cx^2 + cx^2 + cx^2 = 3cx^2
cx^2 = /3

Since P = Nm/V = Nm/ 3V.

pV = 1/3 (Nm) which is also equals to NkT

Thus, 1/2 m = 3/2 kT

Since density = mtotal/V = Nm/V,
p = 1/3 (density).

Question 25

State and explain the equation used to calculate the average translational microscopic kinetic energy of one gas molecule.

Answer 25

= 1/2 m = 1/2 mcrms^2 = 3/2 kT

This equation tells us that temperature is a direct measure of the average microscopic kinetic energy with proportionality constant 3/2 k. 3/2 kT is the microsc9kinetic energy possessed by each mono atomic particle. For non-monoatomic particles, the microscopic kinetic energy will be higher due to rotational and vibrational kinetic energies.