A cup of hot coffee is placed on a metal table in a room. Through which modes of heat transfer does the coffee lose heat to its surroundings?

Introduction / Context:
Heat transfer is the process by which thermal energy moves from a hotter object to a cooler one. There are three primary modes: conduction, convection and radiation. A cup of hot coffee cooling on a table is a common example that involves all three modes. This question asks you to identify which mechanisms are actually at work when the coffee loses heat to the environment.

Given Data / Assumptions:

• A cup of hot coffee is placed on a metal table in a room.
• The coffee is hotter than the air, cup and table around it.
• The cup is in contact with the table and with the surrounding air.
• The system is at ordinary room conditions, not in a vacuum.

Concept / Approach:
Conduction is the transfer of heat through direct contact between particles in a solid or between solids in contact. Convection is the transfer of heat by the movement of fluids (liquids or gases), such as warm air rising. Radiation is the transfer of heat by electromagnetic waves, which does not require a medium. A hot cup of coffee cools by conducting heat through the cup and table, by heating and causing motion in the surrounding air (convection), and by emitting infrared radiation to cooler surroundings.

Step-by-Step Solution:
Step 1: Conduction occurs between the hot coffee and the cup, between the cup and the metal table, and within the materials themselves as thermal energy moves from hotter regions to cooler ones. Step 2: Convection occurs in the air around the cup as the air in contact with the hot surfaces warms, becomes less dense, rises and is replaced by cooler air, setting up convection currents. Step 3: Radiation occurs because any object at a temperature above absolute zero emits thermal (infrared) radiation. The hot coffee and cup radiate energy to the cooler surroundings, including the walls, table and other objects. Step 4: All three modes are simultaneously responsible for the overall cooling of the coffee. Step 5: Therefore, the correct choice must include conduction, convection and radiation together.

Verification / Alternative check:
If conduction were absent, the coffee could not warm the cup or table, but in reality both become warm to the touch. If convection were absent, for example in a vacuum where no air is present, the cooling pattern would be very different and much slower in some arrangements, because there would be no rising warm air. If radiation were absent, the coffee could not emit thermal energy across gaps to distant objects. Experimental studies and everyday experience show that all three mechanisms are at work in this simple situation, which is why textbooks often use cooling drinks as textbook examples of combined heat transfer.

Why Other Options Are Wrong:
Conduction and convection only: This ignores the fact that the hot surfaces also radiate infrared energy to the surroundings.

Conduction and radiation only: This would require no moving air, but in a normal room, air motion and convection currents clearly occur around hot objects.
Convection and radiation only: This would ignore heat conduction from the coffee to the cup and from the cup to the table, which is physically incorrect.

Common Pitfalls:
Students sometimes think of only one dominant mechanism (for example, convection in air) and forget that conduction through solids and radiation from surfaces are always present too. Another mistake is to assume that radiation occurs only at very high temperatures, like in the Sun, but in reality every warm object emits infrared radiation. In most real scenarios, all three modes contribute to heat transfer to some extent.

Final Answer:
A cup of hot coffee on a metal table loses heat by conduction, convection and radiation.