Cell free (CF) massive multiple input multiple output (mMIMO) has been suggested as a key solution to meet the high data rate demands of future wireless communications. Studying the performance of these systems in practical scenarios such as in the presence of hardware impairments is of vital importance. In this paper, we study the effect of power amplifier non-linearity on the uplink and downlink sum-rate of CF mMIMO systems. We derive closed form expressions for the uplink and downlink achievable sum-rates of orthogonal frequency division multiplexing (OFDM) based CF mMIMO systems. Our results show that in the uplink the sum-rate does not increase unlimitedly as the number of access points (AP) increases, being upper bounded, contrarily to the ideal linear case. In fact, the rate of each user is limited by the distortion signal of its power amplifier. However, in the downlink, the sum-rate of the system with non-linear power amplifiers is not bounded and increases unlimitedly with the number of APs. Our results also indicate that for the same signal to distortion ratio (SDR) at the power amplifier output or the same normalized saturation level of the power amplifiers, the relative downlink sum-rate degradation is lower than the relative uplink sum-rate degradation (both with respect to their corresponding values in the ideal linear case). In fact, our results confirm that the user side power amplifier non-linearity has higher impact on the system performance than the power amplifier non-linearity on the AP antennas.