Coffee can have a different temperature depending on the container it is served in.
A temperature gradient is the gradual variation of temperature with distance. The gradient slope is consistent with a material. A gradient is established whenever two materials at different temperatures are in physical contact with each other. The units of measurement for temperature gradients are degrees per unit of distance, such as °F per inch or °C per meter.
Many temperature gradients exist naturally, while others are created. The largest temperature gradient on Earth is the Earth itself. The temperature of the Earth’s core is estimated to be around 9,000 °F (5,000 °C); it is 6,650 °F (3,700 °C) at the boundary between the core and the mantle, while the temperature of the crust is around 200 °F (93 °C). Each layer has a temperature gradient of a different slope depending on the heat conductivity of the layer.
There is no temperature gradient between the Earth and the sun because there is no atmosphere between them. Heat capacity is the ability of a material to retain heat. A vacuum has zero heat capacity.
Convection destroys a thermal gradient. When heating a pan with sauce, the liquid closest to the burner becomes the hottest. When stirred, the hot liquid mixes with the cooler liquid, the heat becomes evenly distributed, and the temperature gradient is eliminated.
If not stirred, convective heat transfer will cause the hot liquid to rise and the cold to fall, and some circulation will occur, although it is not as effective as active stirring. Over time, the conduction forces that transfer heat from the bottom will balance the convective forces that cause the water to circulate. If the heat source is low, circulation will be slow, there may be a steep temperature gradient, and the sauce may burn to the bottom. If the heat is high, the sauce will boil, the convection heat transfer will be high, and the temperature gradient will be close to zero.
Insulation is used to retard heat transfer by placing material with low heat conductivity close to the heat source. Insulation helps maintain the thermal gradient between the insulated object and ambient conditions. Coffee will be hotter in a foam cup than in an aluminum cup because foam conducts heat less quickly. Likewise, the coffee drinker may burn a few fingers when picking up the aluminum cup because the thermal gradient is close to zero and the temperature outside the cup is almost the same as the temperature inside the cup.
To be stable, a thermal gradient must have a constant heat source and an available heat sink. Keeping gradients constant is rarely important, except when carrying out chemical reactions. Many industrial processes require careful heat control. The living cell must also maintain careful heat controls for optimal performance. While scientists understand how the human body as a whole maintains a temperature gradient between its core and the outside world, the options available to individual cells are less clear.