Thanks to its low energy consumption and very long range (up to 50 km in free-space), ultra-narrow-band transmission (UNB) represents a promising alternative to classical technologies used in cellular networks to serve low-throughput wireless sensor networks (WSNs) and the Internet of things (IoT). In UNB, nodes access to the medium by selecting their frequency in a random and continuous way. This randomness leads to new behavior in the interference which has not been theoretically analyzed, when considering the pathloss of nodes randomly deployed around the receiver. In this paper, in order to quantify the system performance, we derive and exploit two theoretical expressions of the outage probability in a UNB based IoT network, accounting for both interference due to the spectral randomness and path loss due to the propagation (with and without Rayleigh fading). This enables us to estimate the network capacity as a function of the path-loss exponent, by determining the maximum number of simultaneous supported nodes. We highlight that the bandwidth should be chosen based on the propagation channel properties.