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Chapter 5: Problem 21

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 because it has twice the mass and requires twice the amount of energy to heat compared to a half full pan. Since the heat transfer rate (power) from the stove to the pan is constant, the time required to heat the full pan of water is double the time required to heat the half full pan.

Step by step solution

01

Understand the heat transfer process

In order to heat the water, energy from the stove is transferred to the water in the form of heat until it reaches the boiling point (100 degrees Celsius or 212 degrees Fahrenheit, for pure water). The heat transfer process mainly occurs through the processes of conduction and convection, where energy (in the form of heat) is transferred from a hotter object to a cooler object.
02

Identify energy needed to heat the water

The energy required to heat the water is measured in Joules (J) and is calculated using the formula: Energy (Q) = mass (m) × specific heat capacity (c) × change in temperature (ΔT) Where: - mass (m) is the total mass of water (in kilograms) - specific heat capacity (c) is the amount of energy it takes to raise the temperature of 1 kg of water by 1 degree Celsius (for water, c = 4,184 J/kg°C) - change in temperature (ΔT) is the difference between the starting temperature of the water and the boiling point.
03

Compare the energy needed to heat half and full pan of water

A full pan of water has twice the mass of a half full pan. Thus, considering the same specific heat capacity and the same change in temperature, the energy needed to heat a full pan of water is double the energy needed to heat a half full pan of water. Let's denote the energy required to heat a half full pan as Q1 and the energy required for a full pan as Q2: Q1 = m1 × c × ΔT Q2 = m2 × c × ΔT Since m2 = 2 × m1 (the mass of the full pan is double the mass of the half full pan), we can find Q2 in terms of Q1: Q2 = 2 × Q1
04

Consider the heat transfer rate

The stove delivers an approximately constant power (energy per unit time) to the pan. This can be written as: Power (P) = Energy / Time Let's denote the time required to heat the half full pan as t1, and the time required for a full pan as t2.
05

Calculate the time to heat half and full pans

Using the power formula and the energy needed to heat both pans, we can solve for t1 and t2: t1 = Q1 / P t2 = Q2 / P = (2 × Q1) / P From this step, we can conclude that t2 = 2 × t1. This means that it takes twice the time to heat a full pan of water compared to a half full pan. The reason why a full pan of water takes longer to reach the boiling point than a pan that is half full is because it has twice the mass, and therefore, requires twice the amount of energy to heat. Since the heat transfer rate (power) from the stove to the pan is constant, the time required to heat the full pan of water is double the time required to heat the half full pan.

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