Heat and Temperature

Thermal equilibrium: two bodies in thermal contact are in thermal equilibrium when there is no net flow of heat between them, which happens when they are at the same temperature.

Application in a mercury thermometer: When the bulb is placed in contact with a body (for example, a patient's mouth), heat flows between the body and the mercury until both reach the same temperature. At that point the mercury and the body are in thermal equilibrium, and the mercury has expanded to a definite length in the capillary. By the zeroth law, the scale reading at this length gives the temperature of the body. The thermometer must therefore be left in contact long enough to reach equilibrium, and its small heat capacity ensures it hardly alters the temperature of the body.

(a) the nature of the material only

The coefficient of linear expansion ($\alpha$) is a characteristic property of the material — copper expands more than steel for the same temperature rise. It does not depend on the original length, the temperature rise or the cross-sectional area, since these appear separately in $L = L_0(1 + \alpha\,\Delta\theta)$.

Heat is the form of energy that flows from a body at higher temperature to a body at lower temperature because of the temperature difference. It is the total thermal energy of a body (the energy transferred between bodies). SI unit: joule (J).

Temperature is the degree of hotness or coldness of a body. It is a measure of the average kinetic energy of the molecules of the body and decides the direction of heat flow. SI unit: kelvin (K).

Differences

Why temperature stays constant during a change of state

During melting or boiling, the heat supplied is used to break or loosen the bonds between molecules (increasing their potential energy) and to do work against the external pressure as the volume changes. It does not increase the average kinetic energy of the molecules. Since temperature measures the average kinetic energy, it stays constant until the change of state is complete. This hidden heat is the latent heat.

Coefficient of linear expansion ($\alpha$)

The coefficient of linear expansion of a solid is defined as the fractional increase in length per unit rise in temperature.

If a rod of length $l$ at temperature $\theta$ expands to length $l'$ at temperature $\theta + \Delta\theta$, then

$\alpha = \frac{l' - l}{l\,\Delta\theta} = \frac{\Delta l}{l\,\Delta\theta}$

Coefficient of superficial expansion ($\beta$)

The coefficient of superficial expansion of a solid is defined as the fractional increase in surface area per unit rise in temperature.

If a solid sheet of surface area $A$ at temperature $\theta$ expands to area $A'$ at temperature $\theta + \Delta\theta$, then

$\beta = \frac{A' - A}{A\,\Delta\theta} = \frac{\Delta A}{A\,\Delta\theta}$

Relation between $\alpha$ and $\beta$

Consider a rectangular sheet of a solid whose length is $l$ and breadth is $b$ at temperature $\theta$. Its surface area at this temperature is

$A = l\,b$

Let the temperature be raised by $\Delta\theta$. If the coefficient of linear expansion of the solid is $\alpha$, the new length and breadth are

$l' = l(1 + \alpha\,\Delta\theta)$ $b' = b(1 + \alpha\,\Delta\theta)$

The new surface area is therefore

$A' = l'\,b' = l(1 + \alpha\,\Delta\theta)\; b(1 + \alpha\,\Delta\theta)$ $A' = lb\,(1 + \alpha\,\Delta\theta)^2 = A\,(1 + \alpha\,\Delta\theta)^2$

Expanding the bracket,

$A' = A\,(1 + 2\alpha\,\Delta\theta + \alpha^2\Delta\theta^2)$

Since $\alpha$ is very small (of the order of $10^{-5}$ per °C), the term $\alpha^2\Delta\theta^2$ is negligibly small and can be dropped. Hence

$A' = A\,(1 + 2\alpha\,\Delta\theta)$ $A' - A = 2\alpha\,A\,\Delta\theta$ $\frac{A' - A}{A\,\Delta\theta} = 2\alpha$

But by definition $\dfrac{A' - A}{A\,\Delta\theta} = \beta$. Hence

$\boxed{\beta = 2\alpha}$

Thus the coefficient of superficial expansion of a solid is twice its coefficient of linear expansion.

Principle of calorimetry:
When two or more bodies at different temperatures are mixed in an isolated system, the total heat lost by the hotter bodies is equal to the total heat gained by the colder bodies, provided no heat is lost to the surroundings.

[ \text{Heat lost} = \text{Heat gained} ]

Given:

• Mass of calorimeter, (m_c = 50,\text{g})
• Specific heat capacity of calorimeter, (c_c = 0.4,\text{J g}^{-1},^{\circ}\text{C}^{-1})
• Mass of water, (m_w = 100,\text{g})
• Initial temperature of water and calorimeter, (\theta_1 = 20^{\circ}\text{C})
• Temperature of steam, (\theta_2 = 100^{\circ}\text{C})
• Final temperature of mixture, (\theta = 60^{\circ}\text{C})
• Specific heat capacity of water, (c_w = 4.2,\text{J g}^{-1},^{\circ}\text{C}^{-1})
• Latent heat of vaporization of steam, (L = 2260,\text{J g}^{-1})

Let the mass of steam condensed be (m_s).

Heat gained by calorimeter and water:

[ Q_{\text{gained}} = (m_c c_c + m_w c_w)(\theta - \theta_1) ]

[ Q_{\text{gained}} = (50 \times 0.4 + 100 \times 4.2)(60 - 20) ]

[ Q_{\text{gained}} = (20 + 420)(40) ]

[ Q_{\text{gained}} = 440 \times 40 = 17600,\text{J} ]

Heat lost by steam:

Steam first condenses at (100^{\circ}\text{C}), then the condensed water cools from (100^{\circ}\text{C}) to (60^{\circ}\text{C}).

[ Q_{\text{lost}} = m_s L + m_s c_w (100 - 60) ]

[ Q_{\text{lost}} = m_s(2260 + 4.2 \times 40) ]

[ Q_{\text{lost}} = m_s(2260 + 168) ]

[ Q_{\text{lost}} = 2428,m_s ]

By the principle of calorimetry:

[ Q_{\text{lost}} = Q_{\text{gained}} ]

[ 2428,m_s = 17600 ]

[ m_s = \frac{17600}{2428} ]

[ m_s = 7.25,\text{g} ]

Answer:
The mass of steam condensed is approximately

[ \boxed{7.25,\text{g}} ]

The triple point of water is the unique condition ($T=273.16\ \text{K}$, $P=611.7\ \text{Pa}$) at which ice, liquid water and water vapour coexist in equilibrium. It occurs at exactly one temperature and one pressure, so it can be reproduced precisely in any laboratory with a sealed triple-point cell. The ice and steam points, by contrast, depend on the pressure and on impurities, so they are less sharp and less reproducible. This makes the triple point a reliable standard fixed point.