Interval scale -- very basic question

[Vital]

Hello!

I marked the thread as a basic high school level, because I assume my question is just at that level. )
I am reading some materials on statistics now, and, not having enough background yet, I stumbled upon this sentence:

"As an example, 50°C, although five times as large a number as 10°C, does not represent five times as much temperature. "

Please, help me to understand what does it mean "does not represent five times as much temperature" if 10 x 5 = 50?

I realize how odd and illiterate this question might sound, but, please, don't judge me to harsh for this. )

Thank you very much.

 

[Stephen Tashi]

Please, help me to understand what does it mean "does not represent five times as much temperature" if 10 x 5 = 50?

The interpretation of that statement is a topic in physics (rather than statistics) and it involves accepting a definition of "temperature" that is different than the definition of "degrees centigrade". If you were to define the temperature of an object to be the measurement of ""degrees centigrade" then, it would be true that a 50 C object has five times the temperature than an object at 10 C.

However, if we define define the temperature of an object to be the mean kinetic energy of its molecules then in an object at 0 C, its molecules are still moving, so the object does not have zero temperature. This shows that "degrees centigrade" is not the same as temperature nor is it directly proportional to it.

As a simpler example, consider "American Wire Gauge" https://en.wikipedia.org/wiki/American_wire_gauge. The general idea is have a scale that tells about the diameter of conductive wires. But a number on the scale is not directly proportional to the wire diameter. In fact, larger gauge numbers indicate smaller diameters.

 

[Vital]

The interpretation of that statement is a topic in physics (rather than statistics) and it involves accepting a definition of "temperature" that is different than the definition of "degrees centigrade". If you were to define the temperature of an object to be the measurement of ""degrees centigrade" then, it would be true that a 50 C object has five times the temperature than an object at 10 C.

However, if we define define the temperature of an object to be the mean kinetic energy of its molecules then in an object at 0 C, its molecules are still moving, so the object does not have zero temperature. This shows that "degrees centigrade" is not the same as temperature nor is it directly proportional to it.

As a simpler example, consider "American Wire Gauge" https://en.wikipedia.org/wiki/American_wire_gauge. The general idea is have a scale that tells about the diameter of conductive wires. But a number on the scale is not directly proportional to the wire diameter. In fact, larger gauge numbers indicate smaller diameters.

Thank you very much. It's really interesting)

 

[vanhees71]

Hello!

I marked the thread as a basic high school level, because I assume my question is just at that level. )
I am reading some materials on statistics now, and, not having enough background yet, I stumbled upon this sentence:

"As an example, 50°C, although five times as large a number as 10°C, does not represent five times as much temperature. "

Please, help me to understand what does it mean "does not represent five times as much temperature" if 10 x 5 = 50?

I realize how odd and illiterate this question might sound, but, please, don't judge me to harsh for this. )

Thank you very much.

It's utter nonsense to try to take multiples of temperatures on the usual centigrade scale. It is an arbitrary men-made measure, defined originally by setting the freezing point and boiling point of water at normal pressure to 0 and 100 degrees. This definition was refined over the history of physics, but it's not important in our context.

Only the absolute temperature, measured in Kelvin provides a scale, where it makes some sense take multiples the intensive quantity temperature, because it is related to energy, an extensive quantity. The Kelvin scale defines the absolute 0 to occur as the temperature for a system in its ground state, i.e., the state of minimum energy. For an ideal monatomic gas the equipartition theorem then tells you that the thermal energy (i.e., the total average kinetic energy of all the gas molecules at the given temperature in the rest frame of its center of mass, the rest frame of the thermal system)
$$U=\frac{3}{2} N k_{\mathrm{B}} T,$$
where $N$ is the total number of particles and $k_{\text{B}}$ the Boltzmann constant which converts the arbitrary centigrade units of temperature to energy units (in the SI Joule). This total average energy is what's called internal energy of an ideal gas in the phenomenological thermodynamics, and it is an extensive quantity, for which it makes sense to say something is some multiple times larger than another.

Now it makes some sense to say the absolute temperature $T_2=5 T_1$, because according to the above consideration
$$u=\frac{U}{N}=\frac{3}{2} k_{\mathrm{B}} T,$$
is the average kinetic energy of a single particle in an ideal gas in thermal equilibrium at temperature $T$. Then to sayd $T_2=5 T_1$ simply means that you make the average kinetic energy of a single particle five times as large by heating up the gas.