Question

10 g samples of three different metals, each initially at 20 °c, are left in the sun for 10 minutes. the temperatures of the samples are then measured at 36 °c, 27 °c, and 41 °c respectively. which one has the highest specific heat?
sample c (41 °c)
sample b (27 °c)
all three samples have the same specific heat.
sample a (36 °c)

Explanation:

Step1: Recall the heat formula

The formula for heat absorbed is \( Q = mc\Delta T \), where \( Q \) is heat, \( m \) is mass, \( c \) is specific heat, and \( \Delta T \) is temperature change.

Step2: Analyze the given conditions

All samples have the same mass (\( m = 10\space g \)) and are exposed to the sun for the same time, so they absorb approximately the same heat (\( Q \) is constant).

Step3: Relate \( c \) and \( \Delta T \)

From \( Q = mc\Delta T \), when \( Q \) and \( m \) are constant, \( c \) is inversely proportional to \( \Delta T \) (i.e., \( c\propto\frac{1}{\Delta T} \)).

Step4: Calculate temperature changes

- For sample A: \( \Delta T_A = 36 - 20 = 16\space^\circ C \)
- For sample B: \( \Delta T_B = 27 - 20 = 7\space^\circ C \)
- For sample C: \( \Delta T_C = 41 - 20 = 21\space^\circ C \)

Step5: Determine the smallest \( \Delta T \)

Among \( 16\space^\circ C \), \( 7\space^\circ C \), and \( 21\space^\circ C \), the smallest \( \Delta T \) is \( 7\space^\circ C \) (sample B). But wait, no—wait, inverse proportionality: smaller \( \Delta T \) means larger \( c \)? Wait, no, wait: \( Q = mc\Delta T \), so \( c=\frac{Q}{m\Delta T} \). So if \( Q \) and \( m \) are constant, larger \( \Delta T \) means smaller \( c \), and smaller \( \Delta T \) means larger \( c \)? Wait, no, wait sample B has \( \Delta T = 7 \), sample A has \( 16 \), sample C has \( 21 \). Wait, but the question is which has the highest specific heat. Wait, maybe I made a mistake. Wait, no—wait, when you leave them in the sun, they absorb heat. So \( Q \) is the same (same time, same sun exposure, same mass? Wait, mass is same (10g). So \( Q \) is proportional to \( c\Delta T \). So if \( Q \) is constant (same heat absorbed), then \( c \) is inversely proportional to \( \Delta T \). So the metal with the smallest temperature change (smallest \( \Delta T \)) has the highest specific heat? Wait, no—wait, no: \( c = \frac{Q}{m\Delta T} \). So if \( Q \) and \( m \) are constant, then lower \( \Delta T \) means higher \( c \). Wait, sample B has \( \Delta T = 7 \), which is the smallest. But wait, the options: sample B is 27°C. Wait, but maybe I messed up. Wait, no—wait, maybe the heat absorbed is not the same? Wait, no, same time in the sun, same mass, so approximately same \( Q \). Wait, but let's re - check. Wait, the formula is \( Q = mc\Delta T \). So if \( Q \) is constant, \( c \) is inversely proportional to \( \Delta T \). So the metal that increases temperature the least (smallest \( \Delta T \)) has the highest specific heat. Wait, sample B has \( \Delta T = 7 \), which is the smallest. But wait, the options: sample B is 27°C. But wait, maybe I made a mistake. Wait, no—wait, the initial temperature is 20°C. So sample B: 27 - 20 = 7, sample A: 36 - 20 = 16, sample C: 41 - 20 = 21. So sample B has the smallest \( \Delta T \), so highest \( c \)? But wait, the options: sample B is an option. But wait, no—wait, maybe I got the inverse wrong. Wait, \( c=\frac{Q}{m\Delta T} \), so if \( Q \) and \( m \) are constant, then as \( \Delta T \) increases, \( c \) decreases. So the metal with the largest \( \Delta T \) has the smallest \( c \), and the metal with the smallest \( \Delta T \) has the largest \( c \). So sample B has \( \Delta T = 7 \) (smallest), so highest \( c \). But wait, the options: sample B is (27°C). But let's check the options again. Wait, maybe I made a mistake in the heat absorption. Wait, maybe the heat absorbed is not the same. Wait, no, same time in the sun, so same amount of heat energy is incident on each sample (assuming same surface area, but the problem says 10g samples, so maybe same surface area? The problem doesn't mention surface area, so we assume \( Q \) is the same. So according to \( Q = mc\Delta T \), \( c \) is inversely proportional to \( \Delta T \). So the smallest \( \Delta T \) gives the largest \( c \). So sample B has \( \Delta T = 7 \), which is the smallest, so highest specific heat. But wait, the options: sample B is one of the options. But wait, let's check the options again. The options are:

- sample C (41°C)
- sample B (27°C)
- All three samples have the same specific heat.
- sample A (36°C)

Wait, maybe I made a mistake. Wait, no—wait, sample B's \( \Delta T \) is 7, sample A's is 16, sample C's is 21. So sample B has the smallest \( \Delta T \), so highest \( c \). So the answer should be sample B (27°C)? But wait, no—wait, maybe the heat is not the same. Wait, no, the problem says "left in the sun for 10 minutes". So they absorb heat from the sun. The amount of heat absorbed depends on the specific heat? No, no—specific heat is a property of the material. The heat absorbed is \( Q = mc\Delta T \). So if two materials have different specific heats, and same mass, and same heat input, their temperature changes will be different. So in this case, same heat input (same time in sun), same mass, so \( Q \) is constant. Therefore, \( c = \frac{Q}{m\Delta T} \), so lower \( \Delta T \) means higher \( c \). So sample B has \( \Delta T = 7 \), which is the lowest, so highest \( c \). So the answer is sample B (27°C).

Wait, but wait, maybe I messed up. Let's re - calculate \( \Delta T \):

Sample A: 36 - 20 = 16

Sample B: 27 - 20 = 7

Sample C: 41 - 20 = 21

So \( \Delta T \) for B is 7, which is the smallest. So \( c \) is highest for B. So the answer is sample B (27°C).

Answer:

sample B (27 °C)