What would the maximum energy density of a power source based on carbon-14

Various ways of generating electricity from small radioactive power sources have been long known.


The Voyager space probes launched in the 1970s use thermoelectric generation from plutonium sources to generate ~100W of power to keep the probes running when they go very far from the sun. Some small devices have used other sources and other conversion mechanisms to prove power for very low power applications [like pacemakers](http://large.stanford.edu/courses/2015/ph241/degraw2/) where long lasting power that doesn't need to be replaced is an advantage.


But recent claims suggest a breakthrough where devices powered by carbon-14 could replace the batteries in mobile phones ([Nano diamond Batteries web page](https://ndb.technology/news/) highlights stories claiming their produces could "revolutionise mobile phone batteries").


These claims seem like uncritical hype but should be easy to check.


Ignoring engineering issues (like conversion efficiency) how does the energy density of a power source based on C-14 compare to a mobile phone battery?


(I'm guessing a simple calculation based on the radioactive properties of C-14 could give a decent answer and that any battery experts can tell us what a Li-ion battery achieves so we can compare them. Also, nuclear power sources are not "batteries" but they are compact power sources so please don't be distracted by the terminology).




The maximum theoretical power density of isotopically pure $\ce{^14C}$ can be estimated from the nuclear decay data of $\ce{^14C}$ as follows.


The half-life of $\ce{^14C}$ is $T\_{1/2}=5.70\cdot 10^3\ \mathrm a$ [1](https://www.researchgate.net/publication/224248235_Survey_on_Energy_Consumption_Entities_on_the_Smartphone_Platform/figures?lo=1). Thus, the corresponding decay constant is
$$\tau=\frac{\ln2}{T\_{1/2}}=3.9\cdot 10^{-12}\ \mathrm s^{-1}$$
and the molar activity is
$$A\_\mathrm M=\tau N\_\mathrm A=2.3\cdot 10^{12}\ \mathrm{Bq\ mol^{-1}}$$
Since the molar mass of $\ce{^14C}$ is $M=14.0\ \mathrm{g\ mol^{-1}}$, the specific activity of $\ce{^14C}$ is
$$a=\frac{A\_\mathrm M}M=1.7\cdot 10^{11}\ \mathrm{Bq\ g^{-1}}$$


The energy of the beta decay of $\ce{^14C}$ is $E=156.48\ \mathrm{keV}$ [1](https://www.researchgate.net/publication/224248235_Survey_on_Energy_Consumption_Entities_on_the_Smartphone_Platform/figures?lo=1). However, the beta spectrum is continuous because the total energy of the decay process is divided between the electron, the antineutrino, and the recoiling nuclide. The mean beta energy only amounts to $E\_\mathrm\beta=49.45\ \mathrm{keV}$ [1](https://www.researchgate.net/publication/224248235_Survey_on_Energy_Consumption_Entities_on_the_Smartphone_Platform/figures?lo=1).


![C-14 beta spectrum](https://i.stack.imgur.com/ApoSn.png)
Since
$$E\_\mathrm\beta=49.45\ \mathrm{keV}=7.923\cdot 10^{-15}\ \mathrm J$$
the power density (so-called isotopic power) of $\ce{^14C}$ is
$$e\_\mathrm\beta=aE\_\mathrm\beta=1.3\cdot 10^{-3}\ \mathrm{W\ g^{-1}}$$


**How does this compare to a typical mobile phone battery?**


An iPhone 14 pro has a Li-ion battery with capacity of about 3,200 mAh. At a voltage of 3.6 V, this gives a capacity of about 11.5 Wh.


It weighs just over 200g. While the power consumption varies a lot depending on what the phone is doing these [have been estimated](https://www.researchgate.net/publication/224248235_Survey_on_Energy_Consumption_Entities_on_the_Smartphone_Platform/figures?lo=1) to peak at over 4W for some other smartphones. We can estimate the average power consumption by noting that a typical iPhone needs to be recharged every one or two days. 11.5 Wh over a 48h time period suggests an average power consumption of about 240mW.


To provide the *average* power (ignoring conversion efficiency and any other engineering issues) this would require about 185g of C-14. But to provide the *peak* power (say 4W) would require more than 3kg of C-14 which is 15 times heavier than the *total* weight of an existing iPhone 14 pro.


This would be truly a revolution in mobile phone batteries, but not the good sort.




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References
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[1](https://www.researchgate.net/publication/224248235_Survey_on_Energy_Consumption_Entities_on_the_Smartphone_Platform/figures?lo=1) [ICRP Publication 107. Nuclear Decay Data for Dosimetric Calculations. *Ann. ICRP* **2008**, *38* (3).](https://www.icrp.org/publication.asp?id=ICRP%20Publication%20107)


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