Chemical Physics

Theoretical Background

Introduction

Heat transfer is the movement of heat from one place to another. When an object is at a different temperature from its surroundings, heat transfer occurs so that the body and the surroundings reach the same temperature at thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature as required by the second law of thermodynamics.

There are three mechanisms of heat transfer (Figure 1):

Radiation: Thermal energy is transferred to or from a body as electromagnetic radiation. All objects above the temprature of absolute zero radiate heat, and no medium is necessary. The frequency of the radiation is determined by Planck's law. The amount of heat radiated from an object is determined by its surface reflectivity, emissivity, surface area, temperature, and geometric orientation with respect to other objects.
Convection: Heat is transferred from one place to another by the bulk movement of matter. It is usually the dominant mechanism of heat transfer in liquids and gases. Free convection occurs when warm, less dense fluid rises, while cooler and denser fluid falls, creating convection currents. Forced convection occurs when the fluid is forced to flow by an external force such as fans, stirrers, or pumps.
Conductance: Heat is also transferred through the microscopic movement and vibrations of molecules and atoms, which transfer their energy to neighboring particles. This is the dominant mechanism of heat transfer in solid materials, but it also happens in fluids.

In this experiment, we will make the assumption that the radiation is negligible. Heat convection may also be neglected when temperature differences are relatively small. Our main concern will be with conduction.

Heat Conduction

The change in internal energy at a given point within the material over time is governed by thermal energy dynamics. This holds true under the condition that the system's volume remains constant, ensuring that the internal energy variation arises solely from changes in thermal energy, without contributions from work associated with volume changes. Therefore, In this experiment, the change in the internal energy at some point in the material in time is given by:

$${\partial U(\vec{x},t)\over \partial t} = {\partial \over \partial t}[\rho C_P T(\vec{x},t)]=\rho C_P {\partial T(\vec{x},t) \over \partial t}$$

where ρ is the material density, and C_P is its heat capacity. According to the first law of thermodynamics, the change in internal energy in a point $ \vec{x} $ must be equal to the total heat flux to that point. The heat flux $ \vec{q} $, is the amount of heat flowing through a surface per unit time, and is measured in units of W⋅m^-2. It is given by Fourier’s law:

$$\vec{q}=-k\vec{\nabla}T$$

where k is the thermal conductivity of the material, measured in units of W⋅m^-1K^-1, and ∇T is the temperature gradient. Qualitatively, Fourier’s law dictates the direction of heat flux – from the hot areas to the cold ones. By Gauss’s divergence theorem, the total flux into a point is given by minus the divergence of the flux in that point. Using it, we can combine eqautions (1) and (2) together to get the heat equation:

$${\partial U(\vec{x},t)\over \partial t} =\rho C_P {\partial T(\vec{x},t) \over \partial t}=-{\nabla \cdot(-k{\nabla}(\vec{x},t))}=k{\nabla^2 T(\vec{x},t)}$$

and therefore:

$${\partial T(\vec{x},t)\over \partial t} =\alpha {\nabla^2 T(\vec{x},t)}$$

where α is the heat diffusivity constant and is given by α = k/ρC_p and measured in units of m²·s^-1,. In the third equality, we assumed that the heat conductivity is independent of space, i.e., a homogeneous media, and we used the fact that the divergence of the gradient is the Laplacian. At steady-state, the time derivative of the temperature is equal to 0. Therefore, at steady-state the temperature profile is a solution to Laplace’s equation:

$${\nabla^2 T}=0$$

The solution in a particular case can be found by using the problem’s boundary conditions. For ideal gases, the heat conductivity constant is given by:

$$k = {1\over 3}\lambda \bar{v} C_V[A] = {25\pi C_V \over 32N_A} {1\over \pi d^2} \sqrt{RT \over \pi m}$$

$$\lambda = {RT\over \sqrt{2} N_A \pi d^2P}$$

where λ is the mean free path, v is the mean speed, C_v is the molar-heat capacity at constant volume of the gas, [A] is the molar concentration of gas A, N_A is Avogadro's number, m is the molar mass of the gas molecule, P is the gas pressure and d is the molecular diameter.

Measuring the Thermal Conductivity Coefficient

The thermal conductivity of gases is usually measured in a cylindrical cell either containing an electrically heated wire coaxially mounted in a tube of comparatively larger diameter (Figure 2), or consisting of two concentric cylinders with the gas occupying the narrow gap between. With accurate knowledge of the apparatus dimensions and electrical properties, absolute measurements of thermal conductivity can be made.

We will use the former system for this experiment. The electrically heated wire has a radius R₁ and a length L, and the tube has an inner radius R₂. The temperature of the tube, T₂, equals room temperature. The temperature of the wire, T₁, is maintained at a value a few degrees higher than room temperature by passing electric current through it. For measurements on gases, it is necessary to limit the temperature difference, ΔT = T₁ - T₂, to few degrees to prevent convection. The thermal conductivity coefficient k obtained with this apparatus fits to an intermediate temperature between T₁ and T₂. Since the temperature dependence of k is small, the error introduced by assuming that k applies to room temperature is not significant.

Schematic of the apparatus to be used in thermal conductivity measurements. The line in the center of the cylinder represents the electrically heated wire.

We will make several assumptions: (i) The system has a cylindrical symmetry, (ii) heat flows only in the radial direction, (iii) the system is in steady state, since the temperature does not change within time, and (iv) energy loss due to end effects is negligible.

Let us define dQ/dt (given in [W]) as the total heat transffered through the surface area at a given time, so that dQ/dt = A · q and:

$${dQ\over dt} = -kA_r{dT\over dr}$$

In cylindrical coordinates the total surface area is A_r = 2πrL. The temperature profile can be calculated from eqaution (5). After substituting into equation (8), we get:

$${dQ\over dt} = 2\pi kL{T_1 - T_2\over ln(R_2/R_1)}$$

The rate of heat flow from the wire to the glass wall will be equal to the rate of dissipation of electrical energy as heat by the wire:

$${dQ\over dt} = RI^2 = VI$$

Here R is a resistance of the wire, I is the current through the wire, and V is the voltage between ends of the wire. We also have to take into account that the resistance of the wire depends on the temperature:

$$R = R_0(1 + \alpha \Delta T)$$

where α is the percentage change in resistivity per unit temperature, which is a property of the material from which the wire is made. R₀ is the resistance at 0°C, and can be calculated according to:

$$R_0 = {\rho_0 L \over S}$$

Gas Viscosity

The thermal conductivity coefficient may be used to calculate the viscosity of a gas:

$$\eta = {2kM\over5C_V}$$

where C_v is the specific-heat capacity at constant volume and M is the molecular mass