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Here, a number of practice problems on heat have been collected and answered. In all practice problems, we used the famous formula $Q=mc\Delta T$ to solve them.

*When heat energy $Q$ causes a change in temperature $\Delta T=T_f-T_i$ in a sample with specific heat capacity $c$ and mass $m$, then we can relate all these physical quantities as following formula \[Q=mc\Delta T=mc(T_f-T_i)\] where $T_f$ and $T_i$ are the initial and final temperatures. *

**Problem (1): 5.0 g of copper was heated from 20°C to 80°C. How much energy was used to heat Cu? (Specific heat capacity of Cu is 0.092 cal/g. °C) **

**Solution**: The energy required to change the temperature of a substance of mass $m$ from initial temperature $T_i$ to final temperature $T_f$ is obtained by formula $Q=mc(T_f-T_i)$, where $c$ is the specific heat of the substance. Thus, we have \begin{align*} Q&=mc\Delta T\\ &= 5\times 0.092\times (80^\circ-20^\circ)\\&= 27.6 \quad {\rm cal} \end{align*}

**Problem (2): How much heat is absorbed by a 20 g granite boulder as energy from the sun causes its temperature to change from 10°C to 29°C? (Specific heat capacity of granite is 0.1 cal/g.°C) **

**Solution**: to raise the temperature of the boulder from 10°C to 29°C, we must add $Q=mc\Delta T$ energy to the granite as below \begin{align*} Q&=mc\Delta T\\ &=20 \times 0.1\times (29^\circ-10^\circ)\\&=38\quad {\rm cal}\end{align*}

In all these example problems, there is no change in the state of the substance. If there were a change in the phase of matter (solid $\Leftrightarrow$ liquid) read the following page to learn more:

__Solved problems on latent heat of fusion__

**Problem (3): How much heat is released when 30 g of water at 96°C cools to 25°C? The specific heat of water is 1 cal/g.°C. **

**Solution**: the amount of energy released is obtained by formula $Q=mc\Delta T$ as below \begin{align*} Q&=mc\Delta T\\&=30\times 1\times (25^\circ-96^\circ)\\&= -2130\quad {\rm cal}\end{align*} The negative shows that the energy released from the water.

**Problem (4): If a 3.1 g ring is heated using 10.0 calories, its temperature rises 17.9°C. Calculate the specific heat capacity of the ring. **

**Solution**: since heat causes change in temperature, so the amount of heat is obtained by formula $Q=mc(T_f-T_i)$. By putting known values into it and solving for unknown value, specific heat of ring is calculated as below, \begin{align*} c&=\frac{Q}{m(T_f-T_i)}\\ \\ &=\frac{10}{3.1\times 17.9^\circ}\\ \\&=0.18\quad {\rm cal/g\cdot ^\circ C}\end{align*}Note that in this problem, the difference between temperatures is given not the initial or final temperatures.

**Problem (5): The temperature of a sample of water increases from 20°C to 46.6°C as it absorbs 5650 calories of heat. What is the mass of the sample? (Specific heat of water is 1.0 cal/g.°C)**

**Solution**: As before, using heat formula and solving for mass $m$, we get \begin{align*} m&=\frac{Q}{c\Delta T}\\\\ &=\frac{5650}{1\times (46.6^\circ-20^\circ)}\\ \\&=212.4\quad {\rm g}\end{align*}

**Problem (6): The temperature of a sample of iron with a mass of 10.0 g changed from 50.4°C to 25.0°C with the release of 47 calories of heat. What is the specific heat of iron?**

**Solution**: __specific heat capacity__ is defined as the amount of heat energy required to change the temperature of a sample with mass $m$ by $\Delta T$. Thus, we have \begin{align*} c&=\frac{Q}{m\Delta T}\\ \\&=\frac{47}{10\times (25^\circ-50.4^\circ)}\\ \\&= 0.185\quad {\rm cal/g\cdot ^\circ \! C}\end{align*} Note that the specific heat of materials is a positive quantity, so if you get a negative, as above, you must pick its absolute value.

**Problem (7): A 4.50 g coin of copper absorbed 54 calories of heat. What was the final temperature of the copper if the initial temperature was 25°C? The specific heat of copper is 0.092 cal/g.°C.**

**Solution**: Let $T_i$ and $T_f$ be the initial and final temperatures of the copper's coin. Again using formula $Q=mc(T_f-T_i)$ and solving for final temperature $T_f$, we have \begin{align*} T_f&=\frac{Q}{mc}+T_i \\ \\ &=\frac{54}{0.092\times 4.5}+25^\circ\\ \\ &=155.43\,{\rm ^\circ C}\end{align*}

**Problem (8): A 155 g sample of an unknown substance was heated from 25°C to 40°C. In the process, the substance absorbed 569 calories of energy. What is the specific heat of the substance? **

**Solution**: In the heat formula $Q=mc\Delta T$, specific heat of any substance is denoted by $c$. Putting known values into this formula and solving for unknown specific heat, we get \begin{align*} c&=\frac{Q}{m\Delta T}\\ \\ &=\frac{569}{155\times (40^\circ-25^\circ)}\\ \\&=0.244\quad {\rm cal/g\cdot^\circ\! C} \end{align*}

**Problem (9): What is the specific heat of an unknown substance if a 2.50 g sample releases 12 calories as its temperature changes from 25°C to 20°C?**

**Solution**: same as above, we have \begin{align*} c&=\frac{Q}{m(T_f-T_i)}\\ \\&=\frac{12}{2.5\times (20^\circ-25^\circ)}\\\\&=0.96\quad {\rm cal/g\cdot ^\circ \! C}\end{align*}

**Problem (10): When 3 kg of water is cooled from 80°C to 10°C, how much heat energy is lost? (specific heat of water is $c_W=4.179\,{\rm J/g\cdot \!^\circ C}$)**

**Solution**: heat has led to a change in temperature so we must use the formula $Q=mc\Delta T$ to find the lost heat as below \begin{align*} Q&=mc(T_f-T_i)\\&=3000\times 4.179\times (10^\circ-80^\circ)\\&=-877590\quad {\rm J} \\ or &=-877.590\quad {\rm kJ}\end{align*} Note that in above the value of specific heat is given in grams but the weight of water is in kilogram, so first convert them into grams or kilograms and then continue to solve the problem. Here, we converted 3 kg to 3000 g.

The negative shows that the heat released from the water.

**Problem (11): How much heat is needed to raise a 0.30 kg piece of aluminum from 30°C to 150°C? ($c_{Al}=0.9\,{\rm J/g\cdot \!^\circ C}$)**

**Solution:** Let $T_f$ and $T_i$ be initial and final temperatures of the aluminum so the required heat is computed as below \begin{align*} Q&=mc(T_f-T_i)\\&=0.3\times 900\times (150^\circ-30^\circ)\\&=-32400\quad {\rm J}\\ or &=-32.4\quad {\rm kJ}\end{align*} Here, we converted specific heat in SI units.

**Problem (12): Calculate the temperature change when:
(a) 10.0 kg of water loses 232 kJ of heat. ($c_W=4.179\,{\rm J/g\cdot \!^\circ C}$)
(b) 1.96 kJ of heat are added to 500 g of copper.($c_{Cu}=0.385\,{\rm J/g\cdot \!^\circ C}$)**

**Solution**: In both parts, we use the heat formula when temperature changes, $Q=mc(T_f-T_i)$.

(a) Substituting known values $m=10\,{\rm kg}$ and $Q=232\,{\rm kJ}$ into above equation and solving for change in temperature $\Delta T=T_f-T_i$, we get \begin{align*} \Delta T&= \frac{Q}{mc}\\ \\&=\frac{-232000}{10\times 4179}\\ \\&=-5.55\,{\rm ^\circ C}\end{align*} Since water loses heat energy (which justify why we inserted a minus sign for Q) so its temperature must be decreases. In above, kJ means 1000 J energy.

(b) Heat added to the water so $Q>0$ must be inserted into the formula, \begin{align*}\Delta T&=\frac{Q}{mc}\\ \\&=\frac{1960}{0.5\times 385}\\ \\&=10.18\,{\rm ^\circ C}\end{align*}

**Problem (13): When heated, the temperature of a water sample increased from 15°C to 39°C. It absorbed 4300 joules of heat. What is the mass of the sample?**

**Solution**: putting known values into the equation $Q=mc(T_f-T_i)$ and solving for unknown mass, we get \begin{align*} m&=\frac{Q}{c(T_f-T_i)}\\ \\ &=\frac{4300}{4179\times (15^\circ-39^\circ)}\\ \\&=0.0428\quad {\rm kg}\\ \\ or &=42.8\quad {\rm g} \end{align*}

**Problem (14): 5.0 g of copper was heated from 20°C to 80°C. How much energy was used to heat Cu?**

**Solution**: the necessary energy is calculated as below, \begin{align*} Q&=mc(T_f-T_i)\\&=5\times 0.385\times (20^\circ-80^\circ)\\&=115.5\quad {\rm J}\end{align*}

**Problem (15): The temperature of a sample of iron with a mass of 10.0 g changed from 50.4°C to 25.0°C with the release of 47 Joules of heat. What is the specific heat of iron?**

**Solution**: the specific heat of iron is determined as below \begin{align*} c&=\frac {Q}{m(T_f-T_i)}\\ \\&=\frac{-47}{10\times (25^\circ-50.4^\circ)}\\ \\&= 0.185\quad {\rm J/g\cdot \! ^\circ C} \end{align*} Since heat is released from the sample so we insert a negative in front of heat to get a positive specific heat of capacity.

**Problem (16): The temperature of a sample of water increases from 20°C to 46.6°C as it absorbs 5650 Joules of heat. What is the mass of the sample? **

**Solution**: known values are $T_i={\rm 20^\circ C}$, $T_f={\rm 46.6^\circ C}$ and $Q=5650\,{\rm J}$. Thus, we have \begin{align*} m&=\frac{Q}{c(T_f-T_i)}\\ \\&=\frac{5650}{4179\times (46.6^\circ-20^\circ)}\\ \\ &=0.0508\quad {\rm kg} \\ \\ or &=50.8\quad {\rm g} \end{align*}

**Author**: Ali Nemati

**Page Created:** 3/9/2021

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