For a common class of slowly varying functions \(f\), this note studies the empirical distributions modulo one of sequences \((f(p_n))\) where \(p_n\) denotes the \(n\)-th prime number. By means of a Kantorovich metric for probability measures on the circle, simple explicit bounds are established for the deviation of these empirical distributions from their limit set, a circle consisting of all rotated copies of a single exponential distribution. Despite their simplicity the established bounds often are sharp. They complement known results in the literature and give rise to several intriguing quantitative questions.