In grinding a steel knife, the metal can get as hot as $400 ^ { \circ } \mathrm { C }$ If the blade has a mass of 80 g, what is the minimum amount of water needed at $20 ^ { \circ } \mathrm { C }$ if the water is to remain liquid and not rise above $100 ^ { \circ } \mathrm { C }$ when the hot blade is quenched in it? The specific heat of the steel is $0.11 \mathrm { cal } / \mathrm { g } \cdot \mathrm { C } ^ { \circ }$ and the specific heat of water is $1.0 \mathrm { cal } / \mathrm { g } \cdot \mathrm { C } ^ { \circ }$

Question

In grinding a steel knife, the metal can get as hot as $400 ^ { \circ } \mathrm { C }$ If the blade has a mass of  80 g, what is the minimum amount of water needed at $20 ^ { \circ } \mathrm { C }$ if the water is to remain liquid and not rise above $100 ^ { \circ } \mathrm { C }$ when the hot blade is quenched in it? The specific heat of the steel is $0.11 \mathrm { cal } / \mathrm { g } \cdot \mathrm { C } ^ { \circ }$ and the specific heat of water is $1.0 \mathrm { cal } / \mathrm { g } \cdot \mathrm { C } ^ { \circ }$

Quizplus · Accepted Answer

To find the minimum amount of water needed, we first calculate the heat lost by the steel as it cools from $400 ^ { \circ } \mathrm { C }$ to $100 ^ { \circ } \mathrm { C }$. This heat will be absorbed by the water, which cannot exceed $100 ^ { \circ } \mathrm { C }$.Heat lost by steel = mass of steel × specific heat of steel × change in temperature= $80 \, \text{g} \times 0.11 \, \text{cal/g}^{\circ}\text{C} \times (400 - 100) ^ { \circ } \mathrm { C }$= $80 \times 0.11 \times 300$= $2640 \, \text{cal}$This heat will be absorbed by the water, raising its temperature to $100 ^ { \circ } \mathrm { C }$ from $20 ^ { \circ } \mathrm { C }$, a change of $80 ^ { \circ } \mathrm { C }$.Using the formula for heat absorbed, $Q = m \times c \times \Delta T$, where $m$ is the mass of water, $c$ is the specific heat of water, and $\Delta T$ is the change in temperature, we solve for $m$:$2640 = m \times 1.0 \times 80$$m = \frac{2640}{80}$$m = 33 \, \text{g}$Therefore, the minimum amount of water needed is 33 g.

In Grinding a Steel Knife, the Metal Can Get as Hot