To find the specific heat of an unknown material given temperature change, mass, and energy applied, we use the following equation:

Plug in known values and solve.

The equation that is needed is:

Plug in known values and solve.

In this question, we're presented with the definition of heat capacity. We're also told that the heat capacity for any given material can be measured at 1) constant volume, or 2) constant pressure. We're being asked to determine whether the constant volume heat capacity is greater than, less than, or equal to the constant pressure heat capacity for the same material.

First, let's revisit the definition of heat capacity. It is the amount of heat that must be added to a material to raise the temperature of that material by .

Next, let's consider what is happening as we add heat to a material at constant volume. When the volume of a material is held constant, that means that it is prevented from expanding as heat is added. Thus, all of the heat being added is resulting directly in an increase in the average kinetic energy of the particles that make up that material. In other words, all of the heat being added results in a temperature increase.

Let's now consider what happens when heat is added to a material at constant pressure. In this situation, the material is allowed to expand as heat is added to it (because volume is not being held constant). Therefore, when heat is added to the material, some of the heat will result in an increased temperature, but some of the heat will also result in an expansion of the material. The consequence of this is that in order to increase the temperature of the material by a given amount at constant pressure, it will require more heat energy than at constant volume.

Again, the main point of this is as follows:

• Constant volume - all of the added heat results in increased temperature
• Constant pressure - some of the added heat results in increased temperature and some of the added heat results in expansion

Since only a fraction of the heat being added at constant pressure will actually result in a temperature increase, we'll have to add more heat in order to increase its temperature by a given amount, when compared to constant volume conditions.

Since the two gases are in an isolated system, we know that the heat lost by one is equal to the heat gained by the other. Furthermore, we know that the two gases will be in thermal equilibrium when they have the same temperature, which will be some value in between their two initial values. Since methanol is at a higher temperature, we can say that it will loose heat and ethanol will gain that same amount of heat:

Substituting in expressions for each of these, we get:

Where the final temperature is equal for both gases:

Now we simply need to rearrange for the final temperature:

We have all of these values, so time to plug and chug:

To find out how much the bar will change in length, we'll need to use the equation for thermal expansion, which is as follows:

If we plug in the values that we know, we can solve for the change in length of the bar.

Now, we just need to add this value to the original length of the bar, and we will have our final answer.

According to the second law of thermodynamics, heat flows from hot to cold. Thus, the heat from her hand flows into the ice cube, causing it to melt.

The efficiency of an ideal heat engine is given by:

, where  and  stand for hot and cold respectively. To use this equation our temperatures must be in Kelvin. Here is the equation to convert from Celsius to Kelvin.

.

The denominator does not change if the temperature is hot or cold, so we need only consider the numerator. Let's compare both calculations:

Hot Day:

Cold Day: 

Comparing these two efficiencies, we see that the cold day is roughly  times as efficient.

Adiabatic means that no heat/energy is transferred. Since the gas is compressed and no heat is transferred, the volume decreases causing the pressure and temperature to rise. 

We know that the expression for average kinetic energy is:

Where average velocity of a gas is:

Plugging this into our expression for kinetic energy, we get:

Plugging in our values, we get:

For gases at equal temperatures, we know that they both have the same average kinetic energy:

Note that  is the molecular weight of each molecule, and  is the average velocity of each molecule. Then rearranging for the ratio of velocities:

Then taking the square root of each side:

Now all we need is the molecular weight of each molecule:

Plugging these in, we get: