Thermodynamics

Temperature measures the average kinetic energy of the particles. The total thermal energy also depends on how many particles there are (the mass), which is why a bathtub of warm water holds more energy than a cup of boiling water.

Heat always flows spontaneously from higher to lower temperature. It continues until the two objects reach the same temperature – thermal equilibrium – when the net flow stops.

Radiation is the transfer of energy by electromagnetic waves and needs no medium, so it crosses empty space. Conduction and convection both need particles, so they cannot occur in a vacuum.

Convection needs a fluid that can flow. When part of a liquid or gas is heated it expands, becomes less dense and rises, while cooler fluid sinks to take its place – setting up a circulating current. Solids can't flow, so they transfer heat by conduction instead.

Thermal equilibrium means equal temperatures, so heat no longer flows on average from one to the other. They need not hold the same amount of energy – a large cool object can store more than a small hot one.

In conduction, energetic particles jostle their neighbours and pass energy along, while staying roughly in place. It works best in solids, and especially in metals, where free electrons carry energy quickly.

$Q = mc\Delta T = (2.0)(4200)(10) = 84\,000\text{ J}$ (that is $84\text{ kJ}$).

A temperature difference is the same number in °C and K, because both scales use the same size degree – they differ only by a fixed offset of $273$. So a rise of $20^\circ\text{C}$ is a rise of $20\text{ K}$. (Adding $273$ to get $293$ would convert an absolute temperature, not a change.)

From $\Delta T = \dfrac{Q}{mc}$, with the same $Q$ and $m$ the temperature rise is inversely proportional to $c$. The oil's specific heat is half the water's, so its temperature rises twice as much. (This is why water, with its high specific heat, makes such an effective coolant.)

$Q = mL = (0.20)(3.3\times10^{5}) = 66\,000\text{ J}$. No temperature change happens – all of this energy goes into breaking the bonds holding the solid together.

Temperature tracks the average kinetic energy, and it stays constant during melting – so the energy is not going into kinetic energy. Instead it breaks the bonds holding the solid together, raising the particles' potential energy. This stored latent heat is released again when the liquid refreezes.

A flat section on a heating curve is a change of state: the substance absorbs latent heat while its temperature stays constant, until the phase change is complete. The two flat sections here are melting and boiling.

$\dfrac{Q}{t} = \dfrac{kA\,\Delta T}{d} = \dfrac{(400)(0.01)(50)}{0.20} = \dfrac{200}{0.20} = 1000\text{ W}$.

Energy is conserved: in an insulated system the heat lost by the hotter object is exactly the heat gained by the cooler one. Setting heat lost = heat gained is the basis of every calorimetry calculation.

A high specific heat means each kilogram of water needs a lot of energy to change its temperature by one degree. Large bodies of water therefore warm and cool slowly, smoothing out the temperature swings of nearby land.

Both are at $100^\circ\text{C}$, but as steam condenses it gives up its large latent heat of vaporisation to the skin before the resulting hot water even begins to cool. That extra burst of energy is what makes steam burns so dangerous.

Two steps: first melt the ice, then warm the water. $Q = mL_f + mc\Delta T = (0.10)(3.3\times10^{5}) + (0.10)(4200)(20) = 33\,000 + 8400 = 41\,400\text{ J}$. (Forgetting the melting step gives $8400$; forgetting the warming step gives $33\,000$.)

Heat lost by the hot water equals heat gained by the cold: $(3.0)(4200)(60 - T) = (1.0)(4200)(T - 20)$, so $3(60 - T) = T - 20$, giving $200 = 4T$ and $T = 50^\circ\text{C}$. The answer sits closer to the larger (hot) mass – not at the simple midpoint of $40^\circ\text{C}$.

The rate is inversely proportional to the thickness $d$ (it sits on the bottom of the fraction). Doubling $d$ therefore halves the rate – which is exactly why thicker insulation slows heat loss.

Fully separating the particles into a gas takes much more energy than just loosening them from a solid into a liquid. For water $L_v \approx 2.3\times10^{6}\text{ J/kg}$ versus $L_f \approx 3.3\times10^{5}\text{ J/kg}$ – about seven times more – so at a steady heating rate the boiling plateau lasts much longer.

Heat lost by metal = heat gained by water: $(0.50)(400)(100 - T) = (0.50)(4200)(T - 20)$, so $200(100 - T) = 2100(T - 20)$, giving $2300T = 62\,000$ and $T \approx 27^\circ\text{C}$. The water's much larger specific heat keeps the final temperature close to the water's starting value, not halfway.

Condense the steam, then cool the water. $Q = mL_v + mc\Delta T = (0.20)(2.3\times10^{6}) + (0.20)(4200)(60) = 460\,000 + 50\,400 = 5.1\times10^{5}\text{ J}$. (Leaving out the condensation step gives $5.0\times10^{4}$; leaving out the cooling gives $4.6\times10^{5}$.)

Since $\dfrac{Q}{t} = \dfrac{kA\,\Delta T}{d}$, the rate is proportional to $A$ and inversely proportional to $d$. Doubling $A$ multiplies the rate by $2$, and halving $d$ multiplies it by another $2$, giving $2 \times 2 = 4$ times the rate.

Dull, dark surfaces are good emitters (and absorbers) of thermal radiation, while shiny, silver surfaces are poor emitters that reflect radiation back. So the matte black can radiates its heat away faster and cools more quickly – the reason vacuum flasks are silvered inside.

The warm water must first melt the ice, then warm the melted ice to the final temperature $T$: $(0.50)(4200)(30 - T) = (0.10)(3.3\times10^{5}) + (0.10)(4200)T$. This gives $63\,000 - 2100T = 33\,000 + 420T$, so $2520T = 30\,000$ and $T \approx 12^\circ\text{C}$. (A check confirms all the ice melts, since the warm water could supply up to $63\,000\text{ J}$ but only $33\,000\text{ J}$ is needed to melt it.)