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?

The heat that must be supplied to a mass $m$ of a substance to raise its temperature by $\Delta T$ is calculated using $Q = m \cdot c \cdot \Delta T$, where $c$ is the specific heat capacity. For water, $c = 4180\ \text{J}/(\text{kg}\cdot\celsius)$. Calculate the heat required to heat $1\ \text{kg}$ of water so that its temperature rises by $10\ \celsius$.

You want to heat water for pasta and are considering a saucepan with $1\ \text{kg}$ of water and a pot with $2\ \text{kg}$, raising the temperature by the same amount in both cases.

1. a) Which of the two requires more heat?
2. b) Justify your answer using the formula $Q = m \cdot c \cdot \Delta T$.

The specific heat capacity of water is $c = 4180\ \text{J}/(\text{kg}\cdot\celsius)$. Explain in your own words what this number means: how many joules are required and to achieve what, exactly?

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.

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 $40\ \celsius$.

A classmate claims: "If I mix a lot of cold water with a little hot water, the mixture can end up hotter than the hot water was at the beginning, because there is much more water."

1. a) Reason, considering which body releases heat and which gains it, between which two values the final temperature of a mixture of two masses of water must always lie.
2. b) Explain then whether the classmate's statement is possible.
3. c) What condition would cause the final temperature to be very close to that of the hot water?