Understand Thermal Energy Transfers - IB Physics

1. A small, hot metal block is placed in contact with a large, cold metal block in a thermally isolated system. Which quantity must be equal for both blocks once the system has reached thermal equilibrium?
   Answer: The final temperature of each block. Thermal equilibrium is defined as the state where there is no net flow of thermal energy between objects in thermal contact. This condition is met when the objects reach the same final temperature. While the heat lost by the hot block equals the heat gained by the cold block (so the magnitudes of their change in internal energy are equal), their final internal energies will be different due to their different masses and materials. Specific heat capacity is an intrinsic property of the material and does not change.
2. The temperature of a black-body radiator is significantly increased. Which row correctly describes the changes to the total power it radiates and the peak wavelength of its emission spectrum?
   Answer: Total power radiated increases; peak wavelength moves to a shorter wavelength. Two laws govern these changes. The Stefan-Boltzmann law ($L = \sigma A T^4$) states that the total power radiated (L) is proportional to the fourth power of the absolute temperature (T). Therefore, increasing T causes L to increase. Wien's displacement law ($\lambda_{max} T = \text{constant}$) states that the peak wavelength of emission is inversely proportional to the absolute temperature. Therefore, increasing T causes $\lambda_{max}$ to decrease, meaning it moves to a shorter wavelength.
3. When a substance melts at a constant temperature, what are the changes in the average kinetic energy and the average potential energy of its molecules?
   Answer: The average kinetic energy remains constant and the average potential energy increases. Temperature is a measure of the average kinetic energy of the molecules. Since the substance melts at a constant temperature, the average kinetic energy of the molecules must remain constant. The energy added to the substance during melting (latent heat of fusion) is used to break the intermolecular bonds, increasing the separation between molecules. This increases the average potential energy of the molecules.
4. A person sits in front of a campfire and feels warmth. What are the two primary mechanisms of heat transfer that cause this sensation?
   Answer: Convection of hot air and radiation from the fire. The fire emits thermal energy primarily as electromagnetic waves (infrared radiation), which travel through the space to the person. This is radiation. The fire also heats the air around it. This hot air becomes less dense and rises, creating convection currents that can also transfer heat to the person. Conduction through the air is a very inefficient process over this distance. Therefore, radiation and convection are the two primary mechanisms.
5. Two thin metal plates are separated by a small air gap of width $d = 2\text{ mm}$. The plates are at $60°C$ and $40°C$ respectively. Heat transfer occurs through conduction in the still air ($k = 0.025 \text{ W/mK}$) and radiation between the plates ($\epsilon = 0.8$ for both surfaces). Which mechanism dominates the heat transfer, and approximately what fraction of the total heat transfer does it represent?
   Answer: Conduction dominates, representing approximately 85% of total heat transfer Conductive heat flux: $q_{cond} = k\frac{\Delta T}{d} = 0.025 \times \frac{20}{0.002} = 250 \text{ W/m}^2$. Radiative heat flux: $q_{rad} = \epsilon\sigma(T_1^4 - T_2^4) = 0.8 \times 5.67 \times 10^{-8} \times (333^4 - 313^4) \approx 0.8 \times 5.67 \times 10^{-8} \times 2.89 \times 10^{10} \approx 131 \text{ W/m}^2$. Total: $250 + 131 = 381 \text{ W/m}^2$. Conduction fraction: $\frac{250}{381} \approx 0.66$ or about 66%. The closest answer is 65%, so conduction dominates with approximately 65% of the total.
6. A 200 g block of a metal with specific heat capacity $c$ absorbs 4000 J of thermal energy, and its temperature increases by 20 K. A 100 g block of a different metal with specific heat capacity $2c$ absorbs 3000 J of thermal energy. What is the temperature increase of the second block?
   Answer: 15 K This is a two-step problem. First, find the value of $c$ for the first metal using $Q = mc\Delta T$. $4000 = (0.200)(c)(20)$, which gives $4000 = 4c$, so $c = 1000$ J kg⁻¹ K⁻¹. The specific heat capacity of the second metal is $2c = 2000$ J kg⁻¹ K⁻¹. Now, use $Q = mc\Delta T$ for the second block: $3000 = (0.100)(2000)(\Delta T)$. This simplifies to $3000 = 200\Delta T$, so $\Delta T = 15$ K. Alternatively, using ratios: $\Delta T_2 / \Delta T_1 = (Q_2/Q_1) \cdot (m_1/m_2) \cdot (c_1/c_2) = (3000/4000) \cdot (200/100) \cdot (c/2c) = (3/4) \cdot (2) \cdot (1/2) = 3/4$. So $\Delta T_2 = (3/4) \cdot 20 = 15$ K.
7. Water is unusual because its solid phase (ice) is less dense than its liquid phase. What is the primary reason for this anomalous property?
   Answer: Water molecules in ice form a crystalline lattice with significant empty space due to hydrogen bonds. The anomalous density of water is due to its molecular structure and hydrogen bonding. In the solid phase (ice), water molecules are held in a rigid, open hexagonal crystalline lattice by hydrogen bonds. This structure keeps the molecules further apart on average than in the liquid phase, where they are disordered and can pack more closely together. This larger volume for the same mass results in lower density for ice.
8. A spherical object at $100°C$ is cooling in a room at $20°C$ primarily through convection. The convective heat transfer coefficient is $h = 25 \text{ W/m}^2\text{K}$. If the sphere's radius is doubled while maintaining the same material and surface properties, how does the initial rate of cooling ($\frac{dT}{dt}$) change?
   Answer: Decreases by a factor of 2 because the surface area to volume ratio decreases The rate of cooling depends on Newton's law of cooling: $\frac{dT}{dt} = -\frac{hA}{mc}(T-T_{ambient})$. For a sphere, $A = 4\pi r^2$ and $m = \frac{4}{3}\pi r^3 \rho$. When radius doubles: new area = $4 \times$ original, new mass = $8 \times$ original. So $\frac{dT}{dt}_{new} = -\frac{h(4A)}{(8m)c}(T-T_{ambient}) = \frac{1}{2} \times \frac{dT}{dt}_{original}$. The cooling rate decreases by a factor of 2 because the surface area to volume ratio decreases.
9. An electric kettle with a power rating of 2.0 kW is used to boil water. After the water reaches 100 °C, it is observed that 150 g of water turns into steam in 3.0 minutes. What is the specific latent heat of vaporization of water calculated from this data?
   Answer: $2.4 \times 10^6$ J kg⁻¹ First, calculate the total energy $Q$ supplied by the kettle. Power $P = 2.0$ kW = 2000 W. Time $t = 3.0$ min = 180 s. Energy $Q = P \times t = 2000 \times 180 = 360000$ J. This energy vaporized a mass $m = 150$ g = 0.150 kg. The specific latent heat of vaporization $L_v$ is given by $Q = mL_v$. Therefore, $L_v = Q/m = 360000 / 0.150 = 2400000$ J kg⁻¹, which is $2.4 \times 10^6$ J kg⁻¹.
10. A window consists of two parallel glass panes separated by an air gap. In winter, the inside surface is at $20°C$ and the outside surface is at $-10°C$. If the air gap width is doubled while keeping the glass thickness constant, which statement best describes the change in heat transfer?
    Answer: Heat transfer rate remains nearly constant because convection currents in the wider gap increase heat transfer While doubling the air gap thickness would decrease conductive heat transfer ($\dot{Q} \propto 1/d$), the wider gap allows for stronger convection currents to develop. In double-pane windows, convection in the air gap often dominates over pure conduction. The enhanced convection in the wider gap approximately compensates for the reduced conduction, keeping the overall heat transfer rate relatively constant. Option A ignores convection, C incorrectly emphasizes radiation which is minimal at these temperatures, and D confuses energy storage with energy transfer rate.
11. A window pane is a single sheet of glass of thickness 5.0 mm and area 2.0 m². On a cold day, the inner surface is at 15 °C and the outer surface is at 5.0 °C. The thermal conductivity of the glass is 0.80 W m⁻¹ K⁻¹. What is the rate of heat loss through the window?
    Answer: 3.2 kW The rate of heat transfer by conduction is given by $\frac{\Delta Q}{\Delta t} = \frac{kA\Delta T}{\Delta x}$. Here, $k = 0.80$ W m⁻¹ K⁻¹, $A = 2.0$ m², $\Delta T = 15 - 5.0 = 10$ K, and $\Delta x = 5.0$ mm = $5.0 \times 10^{-3}$ m. Substituting these values gives: Rate = $\frac{(0.80)(2.0)(10)}{5.0 \times 10^{-3}} = \frac{16}{5.0 \times 10^{-3}} = 3200$ W. This is equal to 3.2 kW.
12. Star Sirius has a peak emission wavelength $\lambda$ and a luminosity $L$. Star Procyon has a peak emission wavelength of $1.5\lambda$. Both stars can be modelled as black bodies and have approximately the same radius. What is the approximate luminosity of Procyon in terms of $L$?
    Answer: 0.20 L First, use Wien's law ($\lambda_{max} T = \text{constant}$) to find the temperature ratio. $T \propto 1/\lambda_{max}$, so $T_{Procyon} / T_{Sirius} = \lambda / (1.5\lambda) = 1/1.5 = 2/3$. Next, use the Stefan-Boltzmann law ($L = \sigma A T^4$). Since the radii are the same, their surface areas $A$ are the same. Thus, $L \propto T^4$. The ratio of luminosities is $L_{Procyon} / L_{Sirius} = (T_{Procyon} / T_{Sirius})^4 = (2/3)^4 = 16/81 \approx 0.1975$. Therefore, the luminosity of Procyon is approximately 0.20 $L$.
13. Two identical rooms are separated by a wall. Room A is at $25°C$ and Room B is at $15°C$. The wall has thermal conductivity $k$ and thickness $d$. If the wall thickness is reduced to $d/2$ while simultaneously doubling the thermal conductivity to $2k$, by what factor does the rate of heat transfer between the rooms change?
    Answer: Increases by a factor of 4 The rate of heat conduction is given by $\dot{Q} = \frac{kA\Delta T}{d}$. Initially: $\dot{Q}_1 = \frac{kA(10)}{d}$. After changes: $\dot{Q}_2 = \frac{(2k)A(10)}{d/2} = \frac{4kA(10)}{d}$. The ratio is $\frac{\dot{Q}_2}{\dot{Q}_1} = 4$. The thermal conductivity doubles (factor of 2) and the thickness halves (factor of 2), giving a combined factor of $2 \times 2 = 4$.
14. A 50 g ice cube at 0 °C is added to 200 g of water at 25.9 °C in a perfectly insulated container. What is the final equilibrium temperature of the mixture?
    Answer: 5.0 °C First, check if all the ice melts. The heat required to melt the ice is $Q_{melt} = m_{ice}L_f = (0.050)(3.3 \times 10^5) = 16500$ J. The maximum heat the water can lose by cooling to 0 °C is $Q_{cool} = m_{water}c_w\Delta T = (0.200)(4200)(25.9 - 0) = 21756$ J. Since $Q_{cool} > Q_{melt}$, all the ice melts and the final temperature $T_f$ is above 0 °C. By conservation of energy: Heat lost by water = Heat to melt ice + Heat to warm melted ice. $m_w c_w (25.9 - T_f) = m_{ice} L_f + m_{ice} c_w (T_f - 0)$. $(0.200)(4200)(25.9 - T_f) = 16500 + (0.050)(4200)(T_f)$. $840(25.9 - T_f) = 16500 + 210T_f$. $21756 - 840T_f = 16500 + 210T_f$. $5256 = 1050T_f$. $T_f = 5.0$ °C.
15. A liquid in an open, insulated container is observed to cool as it evaporates. What is the primary reason for this cooling effect?
    Answer: The molecules with the highest kinetic energy are the most likely to escape. The molecules in a liquid have a distribution of kinetic energies. Evaporation is the process where molecules at the surface with sufficient kinetic energy to overcome intermolecular forces escape into the vapor phase. The molecules that escape are, on average, the most energetic ones. This selective removal of high-energy molecules lowers the average kinetic energy of the molecules remaining in the liquid. Since temperature is a measure of the average kinetic energy, the liquid's temperature decreases.
16. A large mass of water in a lake at 2 °C is compared to a small cup of boiling water at 100 °C. Which statement is correct?
    Answer: The lake water has a lower temperature but a greater internal energy. Temperature is a measure of the average kinetic energy of the molecules. The boiling water (100 °C) is clearly at a higher temperature than the lake water (2 °C). Internal energy is the sum of the total kinetic and potential energies of all the molecules in the substance. Although the molecules in the lake have a lower average kinetic energy, the sheer number of molecules in the large mass of the lake is vastly greater than in the small cup. Consequently, the total internal energy of the lake water is much greater than that of the cup of boiling water.
17. Two stars, X and Y, radiate as black bodies. The radius of star X is one-quarter the radius of star Y ($R_X = 0.25 R_Y$). The total power radiated by star X is 16 times the power radiated by star Y ($L_X = 16 L_Y$). What is the ratio of the surface temperature of X to that of Y ($T_X / T_Y$)?
    Answer: 4 The Stefan-Boltzmann law states that $L = \sigma A T^4$. For a sphere, $A = 4\pi R^2$, so $L = \sigma (4\pi R^2) T^4$. We can form a ratio: $L_X / L_Y = (R_X / R_Y)^2 (T_X / T_Y)^4$. We are given $L_X / L_Y = 16$ and $R_X / R_Y = 0.25$ or $1/4$. Substituting these values: $16 = (0.25)^2 (T_X / T_Y)^4 \Rightarrow 16 = (1/16) (T_X / T_Y)^4$. To solve for the temperature ratio, we rearrange: $(T_X / T_Y)^4 = 16 \times 16 = 256$. Therefore, $T_X / T_Y = \sqrt[4]{256} = 4$.
18. Which of the following is the most fundamental condition required for heat transfer by natural convection to occur within a fluid?
    Answer: The fluid must be in a gravitational field. Natural convection is driven by buoyancy forces that arise from density differences within the fluid. These density differences are typically caused by temperature variations. However, buoyancy force itself (the upward force exerted by a fluid that opposes the weight of an immersed object) only exists in the presence of a gravitational field. Without gravity, the less dense, warmer fluid would not rise, and the denser, cooler fluid would not sink, so no convection currents would form.
19. A metal spoon and a wooden spoon are both at room temperature. The metal spoon feels colder to the touch than the wooden spoon. Which statement best explains this observation?
    Answer: The metal is a better thermal conductor, so it transfers heat from the hand faster. Both spoons are at the same room temperature. The sensation of 'cold' is due to the rate of heat transfer away from your hand. Metal is a much better thermal conductor than wood. When you touch the metal spoon, it conducts heat away from your hand rapidly, creating a cold sensation. Wood, being an insulator, conducts heat away much more slowly.
20. The rate of heat flow by conduction through a solid bar is $P$. The length of the bar is then doubled, and its cross-sectional area is halved. All other factors, including the temperature difference across its ends, remain constant. What is the new rate of heat flow?
    Answer: $P / 4$ The rate of heat flow by conduction $P$ is given by the formula $P = kA\Delta T / L$, where $k$ is the thermal conductivity, $A$ is the cross-sectional area, $\Delta T$ is the temperature difference, and $L$ is the length. The rate of flow is directly proportional to the area and inversely proportional to the length. If the length is doubled ($L \to 2L$) and the area is halved ($A \to A/2$), the new rate $P_{new}$ will be proportional to $(A/2) / (2L) = A / (4L)$. This is one-quarter of the original proportionality, so the new rate of heat flow is $P/4$.

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