Temperature and temperature difference

An electric kettle heats water from the tap temperature of $20\ \celsius$ to the $70\ \celsius$ needed by grandma for her herbal tea. Calculate the temperature change $\Delta T$ of the water.

You leave a cup of coffee at $80\ \celsius$ on a kitchen table that is at $20\ \celsius$, as shown in the figure.

1. a) In which direction does heat flow: from the coffee to the air or from the air to the coffee?
2. b) What temperature does the coffee tend towards if you leave it for a long time?

Aitana leaves the spoon inside the pot of hot soup while she sets the table. When she returns, the spoon is as hot as the soup.

1. a) Explain in your own words what thermal equilibrium is using this example.
2. b) When two bodies are in thermal equilibrium, does heat continue to pass from one to the other?

A heater supplies $83\,600\ \text{J}$ to a certain mass of water, and its temperature rises by $20\ \celsius$.

1. a) Rearrange the formula $Q = m \cdot c \cdot \Delta T$ to make mass the subject.
2. b) Calculate how many kilograms of water have been heated.

In a frying pan there is $1\ \text{kg}$ of oil ($c = 2000\ \text{J}/(\text{kg}\cdot\celsius)$) and in a saucepan, $1\ \text{kg}$ of water ($c = 4180\ \text{J}/(\text{kg}\cdot\celsius)$). Both receive the same amount of heat: $41\,800\ \text{J}$, as shown in the figure.

1. a) Calculate the temperature rise for each.
2. b) Explain using these numbers why the oil in the frying pan heats up much faster than the water in the saucepan.

In a saucepan, you mix $1\ \text{L}$ of water at $80\ \celsius$ with $1\ \text{L}$ of water at $20\ \celsius$, as shown in the figure.

1. a) Calculate the final temperature of the mixture, assuming there are no heat losses.
2. b) Explain why, since there are equal masses of the same substance, the final temperature is exactly the average of the two.

In an insulated flask, you mix $2\ \text{kg}$ of water at $60\ \celsius$ with $1\ \text{kg}$ of water at $30\ \celsius$. The heat released by the hot water is equal to the heat gained by the cold water.

1. a) Set up the energy balance and calculate the final temperature.
2. b) Explain why the result is not $45\ \celsius$, the simple average between $60\ \celsius$ and $30\ \celsius$.

In an insulated flask, you mix $3\ \text{kg}$ of water at $80\ \celsius$ with $1\ \text{kg}$ of water at $20\ \celsius$.

1. a) Set up the balance $Q_{\text{cedido}} = Q_{\text{ganado}}$ using the formula $Q = m \cdot c \cdot \Delta T$.
2. b) Notice that the specific heat capacity $c$ appears on both sides and cancels out: explain why it is not necessary to know it when both masses are of the same substance.
3. c) Calculate the final temperature of the mixture.

You have $2\ \text{kg}$ of very hot water at $80\ \celsius$, and you want to bring it to $50\ \celsius$ by adding cold tap water, which comes out at $20\ \celsius$, as shown in the figure.

1. a) Set up the heat balance by calling the mass of cold water to be added $m$.
2. b) Calculate this mass.
3. c) Check the result by verifying that the heat released by the hot water equals the heat gained by the cold water.

In a gym shower, the hot water comes out at $70\ \celsius$ and the cold water at $20\ \celsius$. You want to fill a $50\ \text{L}$ bucket with water at $40\ \celsius$ to mop the floor, as shown in the figure.

1. a) Set up the heat balance and find the proportion in which you must mix hot and cold water.
2. b) Calculate how many litres from each tap you need for the $50\ \text{L}$.
3. c) Check that the mixture of these litres is indeed at $40\ \celsius$.

To cool down a tea, you pour $0,1\ \text{kg}$ of cold water at $10\ \celsius$ into the cup, which contains $0,3\ \text{kg}$ of tea (practically water) at $80\ \celsius$.

1. a) Set up the heat balance using the data in kilograms.
2. b) Calculate the final temperature of the mixture.
3. c) If it still seems too hot to you, would you add more cold water or wait for the cup to lose heat to the air? Justify which will cool it down faster and why.