Thermal Physics and the Laws of Gases

Introduction to Thermal Energy and Temperature

The Sun serves as the primary source of thermal energy for all living organisms on Earth. In scientific terms, thermal energy is considered the cause, while temperature is defined as the effect. All organisms require a specific temperature range for survival. This concept is visible in everyday life, such as when a container with a steel bottom is placed on an induction stove in a kitchen. Understanding thermal energy and temperature requires moving beyond common perspectives to a scientific framework that covers energy transfer, effects, and thermodynamic principles.

Scientific Definition and Units of Temperature

Temperature is scientifically defined as the degree of hotness of a body. A hotter body possesses a higher temperature than a colder body. Additionally, temperature is a property that determines whether a body is in thermal equilibrium with its surroundings. It can also be defined in terms of the average kinetic energy of the molecules within a substance. Crucially, temperature is the property that determines the direction of the flow of heat. It is a scalar quantity. The SI unit of temperature is the kelvin ($K$). Other commonly utilized units include degree celsius ($°C$) and degree fahrenheit ($°F$).

The Absolute (Kelvin) Scale of Temperature

The temperature measured in relation to absolute zero using the kelvin scale is known as the absolute scale of temperature, or thermodynamic temperature. Each unit on this scale is defined as the fraction of $\frac{1}{273.16}$ part of the thermodynamic temperature of the triple point of water. A temperature difference of $1\,°C$ is exactly equal to a difference of $1\,K$. Zero Kelvin ($0\,K$) represents the absolute scale of temperature of a body and is equivalent to $-273\,°C$.

The mathematical relations between the different temperature scales are as follows: Between Celsius and Kelvin: $K = C + 273$ Between Fahrenheit and Kelvin: $[K] = (F + 460) \times \frac{5}{9}$

Thermal Equilibrium and Heat Transfer

Two or more physical systems or bodies are in thermal equilibrium if there is no net flow of thermal energy between them. Heat energy always flows from one body to another due to a temperature difference. Therefore, if two bodies are in thermal equilibrium, they must be at the same temperature. When two bodies at different temperatures are brought into contact, heat energy transfers from the hot body to the cold body until equilibrium is established. This process results in a temperature rise for the cold body and a temperature decrease for the hot body until they reach a uniform temperature.

Characteristics and Measurement of Thermal Energy

Thermal energy, also known as heat energy or simply heat, is the form of energy transferred between bodies due to a temperature difference. For example, hot milk left on a table cools because energy flows to the environment, while cold water in a bottle warms because energy flows from the environment into the bottle. Heat is the agent that produces the sensation of warmth. The process of heat energy flowing from a higher temperature body to a lower temperature body is called heating. This transmission can occur via conduction, convection, or radiation. Heat is a scalar quantity, and its SI unit of measurement for energy absorbed or evolved is the joule ($J$).

During the process of heat transfer, the colder body is heated and the hotter body is cooled. While this is sometimes termed cooling, the term heating is more common. Key characteristic features of heat energy transfer include:

1. Heat always flows from a system at a higher temperature to one at a lower temperature.

2. The mass of a system remains unaltered when it is heated or cooled.

3. In any exchange of heat, the heat gained by the cold system is equal to the heat lost by the hot system: $\text{Heat gained} = \text{Heat lost}$.

Alternative Units of Heat Energy

While the joule is the SI unit, calorie and kilocalorie are frequently used in other contexts:

• Calorie: Defined as the amount of heat energy required to raise the temperature of $1\,g$ of water through $1\,°C$.

• Kilocalorie: Defined as the amount of heat energy required to raise the temperature of $1\,kg$ of water through $1\,°C$.

Effects of Heat Energy on Substances

When heat energy is supplied to a substance, it typically undergoes one or more of the following changes:

• The temperature of the substance rises.

• The substance may undergo a phase change (e.g., solid to liquid or liquid to gas).

• The substance expands.

The rise in temperature is proportional to the amount of heat energy supplied and also depends on the mass and nature of the substance.

Thermal Expansion in Solids

Supplying heat to a body can increase its dimensions, a phenomenon known as thermal expansion. In solids, atoms gain energy and vibrate more vigorously, forcing them further apart. Expansion in solids is generally smaller than in liquids and gases due to their rigid nature. There are three types of expansion in solids:

1. Linear Expansion: This occurs when the length of a body changes due to temperature. The coefficient of linear expansion ($\alpha_L$) is the ratio of increase in length per degree rise in temperature to its unit length. Its SI unit is $K^{-1}$. The equation is: $\frac{\Delta L}{L_0} = \alpha_L \Delta T$ Where $\Delta L$ is the change in length ($\text{Final length} - \text{Original length}$), $L_0$ is the original length, and $\Delta T$ is the change in temperature.

2. Superficial Expansion: This is the increase in the area of a solid due to heating. The coefficient of superficial expansion ($\alpha_A$) is the ratio of increase in area per degree rise in temperature to its unit area. Its SI unit is $K^{-1}$. The equation is: $\frac{\Delta A}{A_0} = \alpha_A \Delta T$ Where $\Delta A$ is the change in area and $A_0$ is the original area.

3. Cubical Expansion: This is the increase in volume due to heating. The coefficient of cubical expansion ($\alpha_V$) is the ratio of increase in volume per degree rise in temperature to its unit volume. Its SI unit is $K^{-1}$. The equation is: $\frac{\Delta V}{V_0} = \alpha_V \Delta T$ Where $\Delta V$ is the change in volume and $V_0$ is the original volume.

Coefficient of Cubical Expansion for Common Materials

The following table lists the coefficient of cubical expansion for specific materials in units of $K^{-1}$:

• Aluminium: $7 \times 10^{-5}$

• Brass: $6 \times 10^{-5}$

• Glass: $2.5 \times 10^{-5}$

• Water: $20.7 \times 10^{-5}$

• Mercury: $18.2 \times 10^{-5}$

Expansion in Liquids and Gases

Liquids expand more than solids, and gases exhibit the highest expansion of the three states for a given rise in temperature. While expansion coefficients for liquids are independent of temperature, for gases, the values depend on the temperature.

For liquids, expansion must be measured carefully because they require containers. Heat supplied to a liquid-container system is partly used to expand the container and partly used to expand the liquid. This leads to two types of expansion:

1. Real Expansion: The expansion observed if the liquid were heated directly without a container. The coefficient of real expansion is the ratio of the true rise in volume per degree rise in temperature to the unit volume ($K^{-1}$).

2. Apparent Expansion: The expansion observed without accounting for the expansion of the container. The coefficient of apparent expansion is the ratio of the apparent rise in volume per degree rise in temperature to the unit volume ($K^{-1}$).

Experimental Measurement of Real and Apparent Expansion

To determine these expansions, a liquid is poured into a container to level $L_1$. Upon heating, the container expands first, causing the liquid level to drop to $L_2$. As heating continues, the liquid expands more than the container, and the level rises to $L_3$.

• Apparent Expansion = $L_3 - L_1$

• Real Expansion = $L_3 - L_2$

• Note: Real expansion is always greater than apparent expansion (L_3 - L_2 > L_3 - L_1).

Fundamental Laws of Gases

Three fundamental laws describe the relationship between pressure ($P$), volume ($V$), and temperature ($T$) for gases:

1. Boyle's Law: At constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure ($P \propto \frac{1}{V}$). Alternatively, the product of pressure and volume is constant for a perfect gas: $PV = \text{constant}$.

2. Charles's Law (Law of Volume): Formulated by Jacques Charles, this law states that at constant pressure, the volume of a gas is directly proportional to its temperature ($V \propto T$), or $\frac{V}{T} = \text{constant}$.

3. Avogadro's Law: At constant pressure and temperature, the volume of a gas is directly proportional to the number of atoms or molecules ($n$) present ($V \propto n$), or $\frac{V}{n} = \text{constant}$. Avogadro's number ($N_A$) is the total number of atoms per mole, equal to $6.023 \times 10^{23}/mol$.

Classification of Gases: Real and Ideal

Gases are classified based on the interactions of their particles:

• Real Gases: Gases where molecules or atoms interact with a definite intermolecular or interatomic force of attraction. At very high temperatures or low pressures, real gases behave as ideal gases because these forces become negligible.

• Ideal Gases: Often called perfect gases, these are theoretical gases where atoms or molecules do not interact at all. In practice, no gas is truly ideal, but real gases at low pressure or high temperature approximate ideal behavior. Ideal gases strictly obey Boyle's, Charles's, and Avogadro's laws.

The Ideal Gas Equation

The ideal gas equation or "equation of state" relates all properties of an ideal gas. By combining the three fundamental laws: From Boyle's Law: $PV = \text{constant}$ From Charles's Law: $\frac{V}{T} = \text{constant}$ From Avogadro's Law: $\frac{V}{n} = \text{constant}$

The combined law of gases is expressed as: $\frac{PV}{nT} = \text{constant}$

For a gas containing $\mu$ moles, the number of atoms $n = \mu N_A$. Substituting this into the combined law, we use the Boltzmann constant ($k_B = 1.38 \times 10^{-23}\,JK^{-1}$): $\frac{PV}{\mu N_A T} = k_B$ $PV = \mu N_A k_B T$ By defining $\mu N_A k_B = R$, where $R$ is the universal gas constant ($8.31\,J\,mol^{-1}K^{-1}$), we reach the Equation of State: $PV = RT$

Solved Numerical Problems

Example 1: A container with a capacity of $70\,ml$ is filled with liquid up to $50\,ml$ ($L_1$). Upon heating, the level initially falls to $48.5\,ml$ ($L_2$) and then rises to $51.2\,ml$ ($L_3$).

• Apparent expansion = $L_3 - L_1 = 51.2\,ml - 50\,ml = 1.2\,ml$

• Real expansion = $L_3 - L_2 = 51.2\,ml - 48.5\,ml = 2.7\,ml$

Example 2: A gas at constant temperature is compressed to four times its initial pressure ($P_2 = 4P_1$). If the initial volume ($V_1$) is $20\,cc$ ($20\,cm^3$), find the final volume ($V_2$). Using Boyle's Law ($P_1 V_1 = P_2 V_2$): $V_2 = \frac{P_1 \times V_1}{P_2}$ $V_2 = \frac{P \times 20\,cm^3}{4P}$ $V_2 = 5\,cm^3$