Show that, if two copper blocks with different temperatures are placed in contact, then the direction of spontaneous change is toward the equalization of temperatures. Do so by considering the transfer of \(1 \mathrm{~J}\) of energy as heat from one to the other and assessing the sign of the entropy change. Assume that the temperatures of the blocks remain constant.

The Second Law of Thermodynamics is a fundamental principle that provides deep insight into the direction of natural processes. Simply put, it states that the total entropy of an isolated system can never decrease over time. Entropy, often associated with the level of disorder or randomness in a system, effectively measures the number of ways a system can be arranged.

What this law means in a practical sense is that energy spontaneously spreads and disperses if not hindered. A direct consequence of this is that heat will naturally flow from a warmer object to a cooler one, not the other way around, unless external work is done on the system. The Second Law hence dictates that in a natural process, total entropy must either increase or remain the same.

For instance, if you leave a hot cup of coffee in a cool room, the coffee will not stay hot forever; it will cool down because heat flows from the warmer coffee to the cooler surroundings. This is the Second Law at play, as the process increases the room's entropy.

A spontaneous process in thermodynamics is one that occurs naturally without any external force or energy being applied. A fundamental aspect of these processes is that they increase the overall entropy of the universe. This does not mean that the process happens instantly, but rather that the natural tendency is in that direction.

In thermodynamics, a spontaneous process is all about energy distribution. For instance, perfume diffusing in air or an ice cube melting in a glass of warm water are examples of spontaneous processes because they lead to greater disorder and the scattering of energy.

An important point is that spontaneous does not necessarily mean that all parts of the system experience an increase in entropy. Local decreases in entropy can occur as long as the total entropy, which includes the surroundings, increases. That is why ice can form in a freezer—that process decreases entropy locally, but the energy expelled to the kitchen increases the overall entropy.

Calculating the change in entropy for a system where energy is transferred allows us to quantify the spontaneity of processes. In the case of the copper blocks from our exercise, consider energy as heat (\(Q\)) being transferred from a hotter block to a cooler block at a constant temperature.

The change in entropy for the hot block losing energy is calculated using the formula \( \Delta S = -\frac{Q}{T_{hot}} \), where \( T_{hot} \) is the temperature of the hot block. The negative sign reflects a decrease in entropy due to the loss of heat. In contrast, the cold block gains entropy and is represented by \( \Delta S = \frac{Q}{T_{cold}} \), with \( T_{cold} \) being the temperature of the cold block.

When you sum these changes of entropy, the result gives you the total change for the system. If this result is positive, it verifies that the direction of heat transfer or energy flow was indeed spontaneous, moving toward equilibrium. This calculation is powerful because it provides a quantitative assessment of the second law's effect on matter and energy in a system.