Non-orthogonal multiple access (NOMA) is a multiple access technology that can provide efficient spectrum utilization and increased channel capacity on Internet of Things (IoT) networks. One factor affecting NOMA system performance in the IoT network is the allocation of power for each NOMA user. To improve the channel capacity of the system, this paper presents a method for optimizing power allocation for NOMA in an IoT network with a Lagrange multiplier using the Karush-Kuhn-Tucker (KKT) condition. The optimization will result in an optimal power allocation solution to maximize the system channel capacity with the maximum transmit power constraint function and the minimum data rate for NOMA users. The channel capacity achieved by the system was observed in the Nakagami- m fading channel with imperfect successive interference cancellation (imp-SIC). The proposed power allocation method was compared with orthogonal multiple access (OMA) and the conventional NOMA (C-NOMA) power allocation techniques. The results show that the proposed power allocation coefficient optimization solution with the Lagrange multiplier using the KKT conditions significantly increases the channel capacity of the system compared to the OMA and C-NOMA methods.