  We present a study of carbon radio recombination lines towards Cassiopeia A
using LOFAR observations in the frequency range 10-33 MHz. Individual carbon
$\alpha$ lines are detected in absorption against the continuum at frequencies
as low as 16 MHz. Stacking several C$\alpha$ lines we obtain detections in the
11-16 MHz range. These are the highest signal-to-noise measurements at these
frequencies. The peak optical depth of the C$\alpha$ lines changes considerably
over the 11-33 MHz range with the peak optical depth decreasing from
4$\times10^{-3}$ at 33 MHz to 2$\times10^{-3}$ at 11 MHz, while the line width
increases from 20 km s$^{-1}$ to 150 km s$^{-1}$. The combined change in peak
optical depth and line width results in a roughly constant integrated optical
depth. We interpret this as carbon atoms close to local thermodynamic
equilibrium.
  In this work we focus on how the 11-33 MHz carbon radio recombination lines
can be used to determine the gas physical conditions. We find that the ratio of
the carbon radio recombination lines to that of the 158 $\mu$m [CII]
fine-structure line is a good thermometer, while the ratio between low
frequency carbon radio recombination lines provides a good barometer. By
combining the temperature and pressure constraints with those derived from the
line width we are able to constrain the gas properties (electron temperature
and density) and radiation field intensity. Given the 1$\sigma$ uncertainties
in our measurements these are; $T_{e}\approx68$-$98$ K,
$n_{e}\approx0.02$-$0.035$ cm$^{-3}$ and $T_{r,100}\approx1500$-$1650$ K.
Despite challenging RFI and ionospheric conditions, our work demonstrates that
observations of carbon radio recombination lines in the 10-33 MHz range can
provide insight into the gas conditions.
