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Chapter 5: Problem 22

When heating water on a stove, a full pan of water takes longer to reach the boiling point than a pan that is half full. Why?

Short Answer

Expert verified
Answer: A full pan of water takes longer to reach the boiling point than a half-full pan because it has a greater mass, which requires more heat to raise its temperature. The heat transfer from the stove to the water remains constant, so as the mass of the water increases, the time it takes to reach the boiling point also increases.

Step by step solution

01

Define the variables

Before addressing the problem, let's first define the variables involved: - Q: Heat transferred to the water (measured in joules, J) - m: Mass of the water (measured in kilograms, kg) - c: Specific heat capacity of water (~4186 J/(kg*K)) - 螖T: Change in temperature (measured in degrees Celsius or Kelvin, K) - t: Time it takes for the water to reach boiling point (measured in seconds, s)
02

Understand the heat transfer equation

The relationship between the heat transferred to the water, the mass of water, and the change in temperature can be described by the following equation: Q = mc螖T By rearranging this equation, we can find the time it takes for the water to reach boiling point when heated on a stove: t = (Q/m)梅c
03

Compare the time it takes to heat a full pan and a half-full pan of water

When comparing a full pan of water and a half-full pan of water, we can observe that the mass of water in the full pan is twice the mass of water in the half-full pan (m_full = 2m_half). Using the equation from Step 2, we can write the time it takes for the full pan and half-full pan to reach boiling point as: t_full = (Q/m_full)梅c t_half = (Q/m_half)梅c Substituting m_full = 2m_half: t_full = (Q/(2m_half))梅c
04

Analyze and explain the result

By analyzing the equations above, we can see that the time it takes for a full pan of water to reach boiling point is twice as long as the time it takes for a half-full pan of water, assuming that the heat transfer to the water remains constant. In other words, when the mass of water doubles, the time it takes to reach the boiling point also doubles. The reason why a full pan of water takes longer to reach the boiling point than a half-full pan is that it has more mass, which requires more heat to be transferred to raise its temperature. The heat transfer from the stove to the water is constant, so the greater the mass of water, the longer it will take for the entire volume to reach the boiling point.

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