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I am writing the definition of quantities for Newton's Law of Cooling and am not sure what to actually call k$,$ or its units. This is what I have so far

$$\frac{dT}{dt}=-k(T-T_s)$$

Where $\frac{dT}{dt}$ is the rate of cooling measured in degrees Celsius per second $\left[°C\cdot{}s^{-1}\right]$

$T$ is the temperature of the object measured in degrees Celsius $\left[°C\right]$

$T_s$ is the ambient temperature of the objects surrounding measured in degrees Celsius $\left[°C\right]$

$k$ is a constant and has no units? I don't think that's a good enough description

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marked as duplicate by John Rennie, BMS, Brandon Enright, Kyle Kanos, Prahar May 1 '14 at 16:46

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  • $\begingroup$ $T_s$ would have the units $°C$, not $°C·s^{-1}$. $\endgroup$ – LDC3 May 1 '14 at 4:30
  • $\begingroup$ $k$ in thermodynamics (and quantum mechanics, and some other disciplines touched by either) is Boltzmann's constant. However, in this case, it's a confusingly-named placeholder for a characteristic of the system that may or may not be calculated further on. $\endgroup$ – Blackbody Blacklight May 1 '14 at 4:33
  • $\begingroup$ Sorry copy and paste error. So it would be correct to name the k in $P=-kA\frac{dT}{dx}$ as the Boltzmann's constant as well? I have it listed as the thermal conductivity coefficient at the moment $\endgroup$ – user88720 May 1 '14 at 4:35
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    $\begingroup$ @user88720 No, it's not Boltzmann's constant. Nor does it correspond to other variables named $k$ in thermodynamics. $\endgroup$ – Blackbody Blacklight May 1 '14 at 4:37
  • $\begingroup$ I'm confused. When I look at the Wikipedia pages, it states that both thermal conductivity and the Boltzmann constant are denoted with k... $\endgroup$ – user88720 May 1 '14 at 4:40
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$k$ in thermodynamics (and quantum mechanics, and some other disciplines touched by either) is Boltzmann's constant.

Thermal conductivity (units $\frac{\mathrm W}{\mathrm m^2 \cdot \mathrm K}$) is also named $k$, but that's not what you have there either. Refer to that article for help in deriving what your incomplete equation calls $k$.

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  • $\begingroup$ So the k in this is simply an unknown constant and I could simply use x instead? $\endgroup$ – user88720 May 1 '14 at 4:44
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    $\begingroup$ @user88720 Well, the name $x$ tends to emphasize that a variable is supposed to be isolated and calculated. Maybe $c_0$ is a good neutral name. But yeah, such use of $k$ is too confusing. $\endgroup$ – Blackbody Blacklight May 1 '14 at 4:46
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An another form of Newton's law of cooling is:

enter image description here

(Source:B.L.Worsnop and H.T.Flint, Advanced Practical Physics for Students Ninth Edition, Macmillan) So,k in newtons law of cooling is equal to

enter image description here

where K(in upper case)=thermal conductivity of material A=Surface Area exposed, m=mass, s=specific heat of substance, d=thickness of the body. So.k depends on the nature of the material used and the dimensions of the body.

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